# Find Where Increasing/Decreasing f(x)=4x^3+3x^2-6x+1 Find the derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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 .
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 .
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 .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
Set the derivative equal to .
Solve for .
Factor the left side of the equation.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor.
Factor by grouping.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Multiply by .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
Remove unnecessary parentheses.
Divide each term by and simplify.
Divide each term in by .
Simplify .
Cancel the common factor of .
Cancel the common factor.
Divide 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 by adding the exponents.
Move .
Multiply by .
Multiply by .
Rewrite as .
Multiply by .
Subtract from .
Divide by .
Factor by grouping.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Multiply by .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Set the first factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Set the next factor equal to and solve.
Set the next factor equal to .
Subtract from both sides of the equation.
The final solution is all the values that make true.
The values which make the derivative equal to are .
Split into separate intervals around the values that make the derivative or undefined.
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Multiply by .
Simplify by subtracting numbers.
Subtract from .
Subtract from .
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Multiply by .
Simplify by subtracting numbers.
Subtract from .
Subtract from .
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Find Where Increasing/Decreasing f(x)=4x^3+3x^2-6x+1     