Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^3-3x^2+3 , (-1,3)
,
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
Factor out of .
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Step 1.2.2.1
Factor out of .
Step 1.2.2.2
Factor out of .
Step 1.2.2.3
Factor out of .
Step 1.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4
Set equal to .
Step 1.2.5
Set equal to and solve for .
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Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Add to both sides of the equation.
Step 1.2.6
The final solution is all the values that make true.
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
Raising to any positive power yields .
Step 1.4.1.2.1.2
Raising to any positive power yields .
Step 1.4.1.2.1.3
Multiply by .
Step 1.4.1.2.2
Simplify by adding numbers.
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Step 1.4.1.2.2.1
Add and .
Step 1.4.1.2.2.2
Add and .
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
Raise to the power of .
Step 1.4.2.2.1.3
Multiply by .
Step 1.4.2.2.2
Simplify by adding and subtracting.
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Step 1.4.2.2.2.1
Subtract from .
Step 1.4.2.2.2.2
Add and .
Step 1.4.3
List all of the points.
Step 2
Use the first derivative test to determine which points can be maxima or minima.
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Step 2.1
Split into separate intervals around the values that make the first derivative or undefined.
Step 2.2
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 2.2.1
Replace the variable with in the expression.
Step 2.2.2
Simplify the result.
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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 2.2.2.1.3
Multiply by .
Step 2.2.2.2
Add and .
Step 2.2.2.3
The final answer is .
Step 2.3
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 2.3.1
Replace the variable with in the expression.
Step 2.3.2
Simplify the result.
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Step 2.3.2.1
Simplify each term.
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Step 2.3.2.1.1
One to any power is one.
Step 2.3.2.1.2
Multiply by .
Step 2.3.2.1.3
Multiply by .
Step 2.3.2.2
Subtract from .
Step 2.3.2.3
The final answer is .
Step 2.4
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 2.4.1
Replace the variable with in the expression.
Step 2.4.2
Simplify the result.
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Step 2.4.2.1
Simplify each term.
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Step 2.4.2.1.1
Raise to the power of .
Step 2.4.2.1.2
Multiply by .
Step 2.4.2.1.3
Multiply by .
Step 2.4.2.2
Subtract from .
Step 2.4.2.3
The final answer is .
Step 2.5
Since the first derivative changed signs from positive to negative around , then is a local maximum.
is a local maximum
Step 2.6
Since the first derivative changed signs from negative to positive around , then is a local minimum.
is a local minimum
Step 2.7
These are the local extrema for .
is a local maximum
is a local minimum
is a local maximum
is a local minimum
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