Calculus Examples

Find the first derivative of the function.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Find the second derivative of the function.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Divide by .
Set equal to and solve for .
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Set the factor equal to .
Subtract from both sides of the equation.
The solution is the result of and .
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.
Evaluate the second derivative.
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Multiply by .
Subtract from .
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
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.
Evaluate the second derivative.
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Multiply by .
Subtract from .
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
These are the local extrema for .
is a local maximum
is a local minimum
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