# Find the Absolute Max and Min over the Interval f(x)=2+81x-3x^3 , [0,4]

,
Find the first derivative.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , 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 .
Simplify.
Reorder terms.
Set the first derivative equal to zero.
Solve to find the critical points.
Subtract from 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 .
Divide by .
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
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 the right side.
Simplify each term.
Multiply by .
Raise to the power of .
Multiply by .
Subtract from .
Evaluate the function at .
Simplify the right side.
Simplify each term.
Multiply by .
Raising to any positive power yields .
Multiply by .
Evaluate the function at .
Simplify the right side.
Simplify each term.
Multiply by .
Raise to the power of .
Multiply by .