Calculus Examples

Find the Local Maxima and Minima f(x)=(x^2-1)^3
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate using the chain rule, which states that is where and .
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Step 1.1.1
To apply the Chain Rule, set as .
Step 1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3
Replace all occurrences of with .
Step 1.2
Differentiate.
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Step 1.2.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.4
Simplify the expression.
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Step 1.2.4.1
Add and .
Step 1.2.4.2
Multiply by .
Step 1.2.4.3
Reorder the factors of .
Step 2
Find the second derivative of the function.
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Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Product Rule which states that is where and .
Step 2.3
Differentiate using the chain rule, which states that is where and .
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Step 2.3.1
To apply the Chain Rule, set as .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Differentiate.
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Step 2.4.1
By the Sum Rule, the derivative of with respect to is .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.4
Simplify the expression.
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Step 2.4.4.1
Add and .
Step 2.4.4.2
Multiply by .
Step 2.5
Raise to the power of .
Step 2.6
Raise to the power of .
Step 2.7
Use the power rule to combine exponents.
Step 2.8
Add and .
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Multiply by .
Step 2.11
Simplify.
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Step 2.11.1
Apply the distributive property.
Step 2.11.2
Apply the distributive property.
Step 2.11.3
Apply the distributive property.
Step 2.11.4
Combine terms.
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Step 2.11.4.1
Multiply by by adding the exponents.
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Step 2.11.4.1.1
Move .
Step 2.11.4.1.2
Use the power rule to combine exponents.
Step 2.11.4.1.3
Add and .
Step 2.11.4.2
Move to the left of .
Step 2.11.4.3
Multiply by .
Step 2.11.4.4
Multiply by .
Step 2.11.4.5
Move to the left of .
Step 2.11.4.6
Multiply by .
Step 2.11.5
Simplify each term.
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Step 2.11.5.1
Rewrite as .
Step 2.11.5.2
Expand using the FOIL Method.
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Step 2.11.5.2.1
Apply the distributive property.
Step 2.11.5.2.2
Apply the distributive property.
Step 2.11.5.2.3
Apply the distributive property.
Step 2.11.5.3
Simplify and combine like terms.
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Step 2.11.5.3.1
Simplify each term.
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Step 2.11.5.3.1.1
Multiply by by adding the exponents.
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Step 2.11.5.3.1.1.1
Use the power rule to combine exponents.
Step 2.11.5.3.1.1.2
Add and .
Step 2.11.5.3.1.2
Move to the left of .
Step 2.11.5.3.1.3
Rewrite as .
Step 2.11.5.3.1.4
Rewrite as .
Step 2.11.5.3.1.5
Multiply by .
Step 2.11.5.3.2
Subtract from .
Step 2.11.5.4
Apply the distributive property.
Step 2.11.5.5
Simplify.
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Step 2.11.5.5.1
Multiply by .
Step 2.11.5.5.2
Multiply by .
Step 2.11.6
Add and .
Step 2.11.7
Subtract from .
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 using the chain rule, which states that is where and .
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Step 4.1.1.1
To apply the Chain Rule, set as .
Step 4.1.1.2
Differentiate using the Power Rule which states that is where .
Step 4.1.1.3
Replace all occurrences of with .
Step 4.1.2
Differentiate.
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Step 4.1.2.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.4
Simplify the expression.
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Step 4.1.2.4.1
Add and .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.4.3
Reorder the factors of .
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
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.3
Set equal to .
Step 5.4
Set equal to and solve for .
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Step 5.4.1
Set equal to .
Step 5.4.2
Solve for .
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Step 5.4.2.1
Factor the left side of the equation.
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Step 5.4.2.1.1
Rewrite as .
Step 5.4.2.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 5.4.2.1.3
Apply the product rule to .
Step 5.4.2.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.4.2.3
Set equal to and solve for .
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Step 5.4.2.3.1
Set equal to .
Step 5.4.2.3.2
Solve for .
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Step 5.4.2.3.2.1
Set the equal to .
Step 5.4.2.3.2.2
Subtract from both sides of the equation.
Step 5.4.2.4
Set equal to and solve for .
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Step 5.4.2.4.1
Set equal to .
Step 5.4.2.4.2
Solve for .
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Step 5.4.2.4.2.1
Set the equal to .
Step 5.4.2.4.2.2
Add to both sides of the equation.
Step 5.4.2.5
The final solution is all the values that make true.
Step 5.5
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 9.1.3
Raising to any positive power yields .
Step 9.1.4
Multiply by .
Step 9.2
Simplify by adding numbers.
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Step 9.2.1
Add and .
Step 9.2.2
Add and .
Step 10
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 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
Raising to any positive power yields .
Step 11.2.2
Subtract from .
Step 11.2.3
Raise to the power of .
Step 11.2.4
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 13.1.3
Raise to the power of .
Step 13.1.4
Multiply by .
Step 13.2
Simplify by adding and subtracting.
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Step 13.2.1
Subtract from .
Step 13.2.2
Add and .
Step 14
Since there is at least one point with or undefined second derivative, apply the first derivative test.
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Step 14.1
Split into separate intervals around the values that make the first derivative or undefined.
Step 14.2
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 14.2.1
Replace the variable with in the expression.
Step 14.2.2
Simplify the result.
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Step 14.2.2.1
Multiply by .
Step 14.2.2.2
Raise to the power of .
Step 14.2.2.3
Subtract from .
Step 14.2.2.4
Raise to the power of .
Step 14.2.2.5
Multiply by .
Step 14.2.2.6
The final answer is .
Step 14.3
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 14.3.1
Replace the variable with in the expression.
Step 14.3.2
Simplify the result.
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Step 14.3.2.1
Multiply by .
Step 14.3.2.2
Raise to the power of .
Step 14.3.2.3
Subtract from .
Step 14.3.2.4
Raise to the power of .
Step 14.3.2.5
Multiply by .
Step 14.3.2.6
The final answer is .
Step 14.4
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 14.4.1
Replace the variable with in the expression.
Step 14.4.2
Simplify the result.
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Step 14.4.2.1
Multiply by .
Step 14.4.2.2
Raise to the power of .
Step 14.4.2.3
Subtract from .
Step 14.4.2.4
Raise to the power of .
Step 14.4.2.5
Multiply by .
Step 14.4.2.6
The final answer is .
Step 14.5
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 14.5.1
Replace the variable with in the expression.
Step 14.5.2
Simplify the result.
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Step 14.5.2.1
Multiply by .
Step 14.5.2.2
Raise to the power of .
Step 14.5.2.3
Subtract from .
Step 14.5.2.4
Raise to the power of .
Step 14.5.2.5
Multiply by .
Step 14.5.2.6
The final answer is .
Step 14.6
Since the first derivative did not change signs around , this is not a local maximum or minimum.
Not a local maximum or minimum
Step 14.7
Since the first derivative changed signs from negative to positive around , then is a local minimum.
is a local minimum
Step 14.8
Since the first derivative did not change signs around , this is not a local maximum or minimum.
Not a local maximum or minimum
Step 14.9
These are the local extrema for .
is a local minimum
is a local minimum
Step 15