Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^3-27x on -4 , 4
on ,
Step 1
Find the critical points.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Find the first derivative.
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Step 1.1.1.1
Differentiate.
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Step 1.1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2
Evaluate .
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Step 1.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Multiply by .
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
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Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Add to both sides of the equation.
Step 1.2.3
Divide each term in by and simplify.
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Step 1.2.3.1
Divide each term in by .
Step 1.2.3.2
Simplify the left side.
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Step 1.2.3.2.1
Cancel the common factor of .
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Step 1.2.3.2.1.1
Cancel the common factor.
Step 1.2.3.2.1.2
Divide by .
Step 1.2.3.3
Simplify the right side.
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Step 1.2.3.3.1
Divide by .
Step 1.2.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.5
Simplify .
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Step 1.2.5.1
Rewrite as .
Step 1.2.5.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2.6
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.6.1
First, use the positive value of the to find the first solution.
Step 1.2.6.2
Next, use the negative value of the to find the second solution.
Step 1.2.6.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.3
Find the values where the derivative is undefined.
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Step 1.3.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Step 1.4
Evaluate at each value where the derivative is or undefined.
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Step 1.4.1
Evaluate at .
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Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
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Step 1.4.1.2.1
Simplify each term.
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Step 1.4.1.2.1.1
Raise to the power of .
Step 1.4.1.2.1.2
Multiply by .
Step 1.4.1.2.2
Subtract from .
Step 1.4.2
Evaluate at .
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Step 1.4.2.1
Substitute for .
Step 1.4.2.2
Simplify.
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Step 1.4.2.2.1
Simplify each term.
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Step 1.4.2.2.1.1
Raise to the power of .
Step 1.4.2.2.1.2
Multiply by .
Step 1.4.2.2.2
Add and .
Step 1.4.3
List all of the points.
Step 2
Evaluate at the included endpoints.
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Step 2.1
Evaluate at .
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Step 2.1.1
Substitute for .
Step 2.1.2
Simplify.
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Step 2.1.2.1
Simplify each term.
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Step 2.1.2.1.1
Raise to the power of .
Step 2.1.2.1.2
Multiply by .
Step 2.1.2.2
Add and .
Step 2.2
Evaluate at .
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Step 2.2.1
Substitute for .
Step 2.2.2
Simplify.
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Step 2.2.2.1
Simplify each term.
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Step 2.2.2.1.1
Raise to the power of .
Step 2.2.2.1.2
Multiply by .
Step 2.2.2.2
Subtract from .
Step 2.3
List all of the points.
Step 3
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:
Step 4