Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^4-8x^2
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate.
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Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2
Evaluate .
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Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 2
Find the second derivative of the function.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Find the first derivative.
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Step 4.1
Find the first derivative.
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Step 4.1.1
Differentiate.
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Step 4.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.1.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2
Evaluate .
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Step 4.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Multiply by .
Step 4.2
The first derivative of with respect to is .
Step 5
Set the first derivative equal to then solve the equation .
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Step 5.1
Set the first derivative equal to .
Step 5.2
Factor the left side of the equation.
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Step 5.2.1
Factor out of .
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Step 5.2.1.1
Factor out of .
Step 5.2.1.2
Factor out of .
Step 5.2.1.3
Factor out of .
Step 5.2.2
Rewrite as .
Step 5.2.3
Factor.
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Step 5.2.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 5.2.3.2
Remove unnecessary parentheses.
Step 5.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.4
Set equal to .
Step 5.5
Set equal to and solve for .
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Step 5.5.1
Set equal to .
Step 5.5.2
Subtract from both sides of the equation.
Step 5.6
Set equal to and solve for .
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Step 5.6.1
Set equal to .
Step 5.6.2
Add to both sides of the equation.
Step 5.7
The final solution is all the values that make true.
Step 6
Find the values where the derivative is undefined.
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Step 6.1
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 7
Critical points to evaluate.
Step 8
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 9
Evaluate the second derivative.
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Step 9.1
Simplify each term.
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Step 9.1.1
Raising to any positive power yields .
Step 9.1.2
Multiply by .
Step 9.2
Subtract from .
Step 10
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 11
Find the y-value when .
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Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
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Step 11.2.1
Simplify each term.
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Step 11.2.1.1
Raising to any positive power yields .
Step 11.2.1.2
Raising to any positive power yields .
Step 11.2.1.3
Multiply by .
Step 11.2.2
Add and .
Step 11.2.3
The final answer is .
Step 12
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 13
Evaluate the second derivative.
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Step 13.1
Simplify each term.
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Step 13.1.1
Raise to the power of .
Step 13.1.2
Multiply by .
Step 13.2
Subtract from .
Step 14
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 15
Find the y-value when .
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Step 15.1
Replace the variable with in the expression.
Step 15.2
Simplify the result.
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Step 15.2.1
Simplify each term.
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Step 15.2.1.1
Raise to the power of .
Step 15.2.1.2
Raise to the power of .
Step 15.2.1.3
Multiply by .
Step 15.2.2
Subtract from .
Step 15.2.3
The final answer is .
Step 16
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 17
Evaluate the second derivative.
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Step 17.1
Simplify each term.
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Step 17.1.1
Raise to the power of .
Step 17.1.2
Multiply by .
Step 17.2
Subtract from .
Step 18
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 19
Find the y-value when .
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Step 19.1
Replace the variable with in the expression.
Step 19.2
Simplify the result.
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Step 19.2.1
Simplify each term.
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Step 19.2.1.1
Raise to the power of .
Step 19.2.1.2
Raise to the power of .
Step 19.2.1.3
Multiply by .
Step 19.2.2
Subtract from .
Step 19.2.3
The final answer is .
Step 20
These are the local extrema for .
is a local maxima
is a local minima
is a local minima
Step 21