# Find the Critical Points g(y)=(y-2)/((y^2-2y+4)^2) Find the derivative.
Differentiate using the Quotient Rule which states that is where and .
Differentiate.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Simplify the expression.
Multiply by .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Simplify with factoring out.
Multiply by .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
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 .
Since is constant with respect to , the derivative of with respect to is .
Simplify.
Apply the distributive property.
Simplify the numerator.
Simplify each term.
Multiply by .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Subtract from .
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Factor out of .
Rewrite as .
Move the negative in front of the fraction.
Set the derivative equal to .
Solve for .
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The factors for are , which is multiplied by itself times.
occurs times.
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Multiply each term by and simplify.
Multiply each term in by in order to remove all the denominators from the equation.
Simplify .
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
Multiply by .
Simplify .
Use the Multinomial Theorem.
Simplify terms.
Simplify each term.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Multiply by by adding the exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Use the power rule to combine exponents.
Raise to the power of .
Multiply by .
Apply the product rule to .
Raise to the power of .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Multiply by .
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Multiply by .
Multiply by .
Raise to the power of .
Multiply by .
Multiply by .
Raise to the power of .
Multiply by .
Raise to the power of .
Simplify by adding terms.
Subtract from .
Multiply by .
Solve the equation.
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Factor out of .
Factor out of .
Multiply each term in by
Multiply each term in by .
Simplify .
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
Multiply by .
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
The final answer is the combination of both solutions.
Substitute the values of which cause the derivative to be into the original function.
Evaluate.
Simplify the numerator.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Rewrite in a factored form.
Multiply by .
Subtract from .
Simplify the denominator.
Apply the product rule to .
Raise to the power of .
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
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.
Combine and .
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 .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Rewrite in a factored form.
Apply the distributive property.
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Subtract from .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply and .
Expand the denominator using the FOIL method.
Simplify.
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify terms.
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Rewrite as .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Move to the left of .
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Simplify each term.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
Simplify terms.
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Simplify the numerator.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Rewrite in a factored form.
Multiply by .
Subtract from .
Simplify the denominator.
Apply the product rule to .
Raise to the power of .
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Subtract from .
Combine and .
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 .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Rewrite in a factored form.
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply and .
Expand the denominator using the FOIL method.
Simplify.
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify terms.
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Rewrite as .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Simplify each term.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
Subtract from .
Simplify terms.
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Find the domain of the derivative.
Set the denominator in equal to to find where the expression is undefined.
Solve for .
Take the cube root of each side of the equation to set up the solution for
Remove the perfect root factor under the radical to solve for .
Simplify .
Rewrite as .
Pull terms out from under the radical, assuming real numbers.
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Move to the left of .
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Move to the left of .
Multiply by .
Simplify .
Change the to .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Move to the left of .
Multiply by .
Simplify .
Change the to .
The final answer is the combination of both solutions.
The domain is all real numbers.
Interval Notation:
Set-Builder Notation:
Interval Notation:
Set-Builder Notation:
Since there are no values of where the derivative is undefined, there are no additional critical points.
Find the Critical Points g(y)=(y-2)/((y^2-2y+4)^2)   ## Download our App from the store

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