,

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 .

Add and .

Set the first derivative equal to zero.

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 using the AC method.

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Write the factored form using these integers.

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 .

Multiply by .

Multiply by .

Subtract from .

Divide by .

Factor using the AC method.

Write the factored form using these integers.

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.

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.

Use the endpoints and all critical points on the interval to test for any absolute extrema over the given interval.

Evaluate the function at .

Simplify each term.

Multiply by by adding the exponents.

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Raise to the power of .

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Subtract from .

Add and .

Evaluate the function at .

Simplify each term.

Raise to the power of .

Multiply by .

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Add and .

Add and .

Evaluate the function at .

Simplify each term.

Raise to the power of .

Multiply by .

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Add and .

Add and .

Evaluate the function at .

Simplify each term.

Raise to the power of .

Multiply by .

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Subtract from .

Subtract from .

Add and .

Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.

Absolute Maximum:

Absolute Minimum:

Find the Absolute Max and Min over the Interval f(x)=2x^3-3x^2-12x+1 , [-2,3]