Find the Local Maxima and Minima f(x)=x^4+x^3-6x^2

Math
Find the first derivative of the function.
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Differentiate.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Differentiate using the Power Rule which states that is where .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Find the second derivative of the function.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Set equal to and solve for .
Set equal to and solve for .
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Set the factor equal to .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Multiply by .
Factor out of .
Multiply by .
Multiply by .
Simplify the expression to solve for the portion of the .
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Multiply by .
Factor out of .
Multiply by .
Multiply by .
Change the to .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Simplify the expression to solve for the portion of the .
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Multiply by .
Factor out of .
Multiply by .
Multiply by .
Change the to .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
The final answer is the combination of both solutions.
The solution is the result of and .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Raising to any positive power yields .
Multiply by .
Multiply by .
Simplify by adding zeros.
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Add and .
Subtract from .
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Find the y-value when .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raising to any positive power yields .
Raising to any positive power yields .
Raising to any positive power yields .
Multiply by .
Simplify by adding zeros.
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Add and .
Add and .
The final answer is .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Use the power rule to distribute the exponent.
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Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Cancel the common factor of .
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Add and .
Subtract from .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Move the leading negative in into the numerator.
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Add and .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Find the y-value when .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Use the power rule to distribute the exponent.
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Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
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Raise to the power of .
Raise to the power of .
Multiply by .
Multiply by .
Raise to the power of .
Multiply by .
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Add and .
Add and .
Subtract from .
Cancel the common factor of and .
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Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
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Raise to the power of .
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Add and .
Subtract from .
Cancel the common factor of and .
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Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Cancel the common factor of .
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Add and .
Subtract from .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Add and .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps…
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Add and .
Simplify with factoring out.
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Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
The final answer is .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Cancel the common factor of .
Tap for more steps…
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Move to the left of .
Combine using the product rule for radicals.
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Add and .
Add and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Move the leading negative in into the numerator.
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps…
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Subtract from .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Multiply by .
Subtract from .
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Find the y-value when .
Tap for more steps…
Replace the variable with in the expression.
Simplify the result.
Tap for more steps…
Simplify each term.
Tap for more steps…
Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
Tap for more steps…
Raise to the power of .
Raise to the power of .
Multiply by .
Raise to the power of .
Multiply by .
Rewrite as .
Tap for more steps…
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Multiply by .
Rewrite as .
Raise to the power of .
Rewrite as .
Tap for more steps…
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Rewrite as .
Tap for more steps…
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
Tap for more steps…
Factor out of .
Cancel the common factors.
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Add and .
Add and .
Add and .
Cancel the common factor of and .
Tap for more steps…
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
Tap for more steps…
Raise to the power of .
Multiply by by adding the exponents.
Tap for more steps…
Multiply by .
Tap for more steps…
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
Multiply by .
Rewrite as .
Tap for more steps…
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Tap for more steps…
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Rewrite as .
Raise to the power of .
Rewrite as .
Tap for more steps…
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Add and .
Add and .
Cancel the common factor of and .
Tap for more steps…
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Cancel the common factor of .
Tap for more steps…
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Rewrite as .
Expand using the FOIL Method.
Tap for more steps…
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Tap for more steps…
Simplify each term.
Tap for more steps…
Multiply by .
Move to the left of .
Combine using the product rule for radicals.
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Add and .
Add and .
Cancel the common factor of and .
Tap for more steps…
Factor out of .
Cancel the common factors.
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps…
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Subtract from .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps…
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Subtract from .
Simplify with factoring out.
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Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
The final answer is .
These are the local extrema for .
is a local maxima
is a local minima
is a local minima
Find the Local Maxima and Minima f(x)=x^4+x^3-6x^2

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