Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^2-x-6 on 0 , 6
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 1.1.1.3
Differentiate using the Constant Rule.
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Step 1.1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.2
Add and .
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 1.3
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
Apply the product rule to .
Step 1.4.1.2.1.2
One to any power is one.
Step 1.4.1.2.1.3
Raise to the power of .
Step 1.4.1.2.2
Find the common denominator.
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Step 1.4.1.2.2.1
Multiply by .
Step 1.4.1.2.2.2
Multiply by .
Step 1.4.1.2.2.3
Write as a fraction with denominator .
Step 1.4.1.2.2.4
Multiply by .
Step 1.4.1.2.2.5
Multiply by .
Step 1.4.1.2.2.6
Multiply by .
Step 1.4.1.2.3
Combine the numerators over the common denominator.
Step 1.4.1.2.4
Simplify the expression.
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Step 1.4.1.2.4.1
Multiply by .
Step 1.4.1.2.4.2
Subtract from .
Step 1.4.1.2.4.3
Subtract from .
Step 1.4.1.2.4.4
Move the negative in front of the fraction.
Step 1.4.2
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
Raising to any positive power yields .
Step 2.1.2.1.2
Multiply by .
Step 2.1.2.2
Simplify by adding and subtracting.
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Step 2.1.2.2.1
Add and .
Step 2.1.2.2.2
Subtract from .
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
Simplify by subtracting numbers.
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Step 2.2.2.2.1
Subtract from .
Step 2.2.2.2.2
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