Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=(x^3)/3-2x^2+4 , [-2,1]
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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
By the Sum Rule, the derivative of with respect to is .
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
Combine and .
Step 1.1.1.2.4
Combine and .
Step 1.1.1.2.5
Cancel the common factor of .
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Step 1.1.1.2.5.1
Cancel the common factor.
Step 1.1.1.2.5.2
Divide by .
Step 1.1.1.3
Evaluate .
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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
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.3
Multiply by .
Step 1.1.1.4
Differentiate using the Constant Rule.
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Step 1.1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.4.2
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
Divide by .
Step 1.4.1.2.1.3
Raising to any positive power yields .
Step 1.4.1.2.1.4
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
Find the common denominator.
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Step 1.4.2.2.1.1
Write as a fraction with denominator .
Step 1.4.2.2.1.2
Multiply by .
Step 1.4.2.2.1.3
Multiply by .
Step 1.4.2.2.1.4
Write as a fraction with denominator .
Step 1.4.2.2.1.5
Multiply by .
Step 1.4.2.2.1.6
Multiply by .
Step 1.4.2.2.2
Combine the numerators over the common denominator.
Step 1.4.2.2.3
Simplify each term.
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Step 1.4.2.2.3.1
Raise to the power of .
Step 1.4.2.2.3.2
Raise to the power of .
Step 1.4.2.2.3.3
Multiply .
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Step 1.4.2.2.3.3.1
Multiply by .
Step 1.4.2.2.3.3.2
Multiply by .
Step 1.4.2.2.3.4
Multiply by .
Step 1.4.2.2.4
Simplify the expression.
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Step 1.4.2.2.4.1
Subtract from .
Step 1.4.2.2.4.2
Add and .
Step 1.4.2.2.4.3
Move the negative in front of the fraction.
Step 1.4.3
List all of the points.
Step 2
Exclude the points that are not on the interval.
Step 3
Evaluate at the included endpoints.
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Step 3.1
Evaluate at .
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Step 3.1.1
Substitute for .
Step 3.1.2
Simplify.
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Step 3.1.2.1
Simplify each term.
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Step 3.1.2.1.1
Raise to the power of .
Step 3.1.2.1.2
Move the negative in front of the fraction.
Step 3.1.2.1.3
Multiply by by adding the exponents.
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Step 3.1.2.1.3.1
Multiply by .
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Step 3.1.2.1.3.1.1
Raise to the power of .
Step 3.1.2.1.3.1.2
Use the power rule to combine exponents.
Step 3.1.2.1.3.2
Add and .
Step 3.1.2.1.4
Raise to the power of .
Step 3.1.2.2
Find the common denominator.
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Step 3.1.2.2.1
Write as a fraction with denominator .
Step 3.1.2.2.2
Multiply by .
Step 3.1.2.2.3
Multiply by .
Step 3.1.2.2.4
Write as a fraction with denominator .
Step 3.1.2.2.5
Multiply by .
Step 3.1.2.2.6
Multiply by .
Step 3.1.2.3
Combine the numerators over the common denominator.
Step 3.1.2.4
Simplify each term.
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Step 3.1.2.4.1
Multiply by .
Step 3.1.2.4.2
Multiply by .
Step 3.1.2.5
Simplify the expression.
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Step 3.1.2.5.1
Subtract from .
Step 3.1.2.5.2
Add and .
Step 3.1.2.5.3
Move the negative in front of the fraction.
Step 3.2
Evaluate at .
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Step 3.2.1
Substitute for .
Step 3.2.2
Simplify.
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Step 3.2.2.1
Simplify each term.
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Step 3.2.2.1.1
One to any power is one.
Step 3.2.2.1.2
One to any power is one.
Step 3.2.2.1.3
Multiply by .
Step 3.2.2.2
Find the common denominator.
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Step 3.2.2.2.1
Write as a fraction with denominator .
Step 3.2.2.2.2
Multiply by .
Step 3.2.2.2.3
Multiply by .
Step 3.2.2.2.4
Write as a fraction with denominator .
Step 3.2.2.2.5
Multiply by .
Step 3.2.2.2.6
Multiply by .
Step 3.2.2.3
Combine the numerators over the common denominator.
Step 3.2.2.4
Simplify each term.
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Step 3.2.2.4.1
Multiply by .
Step 3.2.2.4.2
Multiply by .
Step 3.2.2.5
Simplify by adding and subtracting.
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Step 3.2.2.5.1
Subtract from .
Step 3.2.2.5.2
Add and .
Step 3.3
List all of the points.
Step 4
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 5