Calculus Examples

Find the Local Maxima and Minima y=x^4-2x^2
Step 1
Write as a function.
Step 2
Find the first derivative of the function.
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Step 2.1
Differentiate.
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Step 2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
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 3
Find the second derivative of the function.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
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Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Multiply by .
Step 3.3
Evaluate .
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Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Multiply by .
Step 4
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 5
Find the first derivative.
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Step 5.1
Find the first derivative.
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Step 5.1.1
Differentiate.
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Step 5.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 5.1.1.2
Differentiate using the Power Rule which states that is where .
Step 5.1.2
Evaluate .
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Step 5.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.2.2
Differentiate using the Power Rule which states that is where .
Step 5.1.2.3
Multiply by .
Step 5.2
The first derivative of with respect to is .
Step 6
Set the first derivative equal to then solve the equation .
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Step 6.1
Set the first derivative equal to .
Step 6.2
Factor the left side of the equation.
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Step 6.2.1
Factor out of .
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Step 6.2.1.1
Factor out of .
Step 6.2.1.2
Factor out of .
Step 6.2.1.3
Factor out of .
Step 6.2.2
Rewrite as .
Step 6.2.3
Factor.
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Step 6.2.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 6.2.3.2
Remove unnecessary parentheses.
Step 6.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 6.4
Set equal to .
Step 6.5
Set equal to and solve for .
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Step 6.5.1
Set equal to .
Step 6.5.2
Subtract from both sides of the equation.
Step 6.6
Set equal to and solve for .
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Step 6.6.1
Set equal to .
Step 6.6.2
Add to both sides of the equation.
Step 6.7
The final solution is all the values that make true.
Step 7
Find the values where the derivative is undefined.
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Step 7.1
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 8
Critical points to evaluate.
Step 9
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 10
Evaluate the second derivative.
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Step 10.1
Simplify each term.
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Step 10.1.1
Raising to any positive power yields .
Step 10.1.2
Multiply by .
Step 10.2
Subtract from .
Step 11
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 12
Find the y-value when .
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Step 12.1
Replace the variable with in the expression.
Step 12.2
Simplify the result.
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Step 12.2.1
Simplify each term.
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Step 12.2.1.1
Raising to any positive power yields .
Step 12.2.1.2
Raising to any positive power yields .
Step 12.2.1.3
Multiply by .
Step 12.2.2
Add and .
Step 12.2.3
The final answer is .
Step 13
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 14
Evaluate the second derivative.
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Step 14.1
Simplify each term.
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Step 14.1.1
Raise to the power of .
Step 14.1.2
Multiply by .
Step 14.2
Subtract from .
Step 15
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 16
Find the y-value when .
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Step 16.1
Replace the variable with in the expression.
Step 16.2
Simplify the result.
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Step 16.2.1
Simplify each term.
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Step 16.2.1.1
Raise to the power of .
Step 16.2.1.2
Raise to the power of .
Step 16.2.1.3
Multiply by .
Step 16.2.2
Subtract from .
Step 16.2.3
The final answer is .
Step 17
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 18
Evaluate the second derivative.
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Step 18.1
Simplify each term.
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Step 18.1.1
One to any power is one.
Step 18.1.2
Multiply by .
Step 18.2
Subtract from .
Step 19
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 20
Find the y-value when .
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Step 20.1
Replace the variable with in the expression.
Step 20.2
Simplify the result.
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Step 20.2.1
Simplify each term.
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Step 20.2.1.1
One to any power is one.
Step 20.2.1.2
One to any power is one.
Step 20.2.1.3
Multiply by .
Step 20.2.2
Subtract from .
Step 20.2.3
The final answer is .
Step 21
These are the local extrema for .
is a local maxima
is a local minima
is a local minima
Step 22