Calculus Examples

Find the Local Maxima and Minima x^3-12x
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
Differentiate using the Constant Rule.
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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
Add and .
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
Add to both sides of the equation.
Step 6.3
Divide each term in by and simplify.
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Step 6.3.1
Divide each term in by .
Step 6.3.2
Simplify the left side.
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Step 6.3.2.1
Cancel the common factor of .
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Step 6.3.2.1.1
Cancel the common factor.
Step 6.3.2.1.2
Divide by .
Step 6.3.3
Simplify the right side.
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Step 6.3.3.1
Divide by .
Step 6.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6.5
Simplify .
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Step 6.5.1
Rewrite as .
Step 6.5.2
Pull terms out from under the radical, assuming positive real numbers.
Step 6.6
The complete solution is the result of both the positive and negative portions of the solution.
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Step 6.6.1
First, use the positive value of the to find the first solution.
Step 6.6.2
Next, use the negative value of the to find the second solution.
Step 6.6.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Multiply by .
Step 11
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 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
Raise to the power of .
Step 12.2.1.2
Multiply by .
Step 12.2.2
Subtract from .
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
Multiply by .
Step 15
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 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
Multiply by .
Step 16.2.2
Add and .
Step 16.2.3
The final answer is .
Step 17
These are the local extrema for .
is a local minima
is a local maxima
Step 18