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Calculus Examples
Step 1
Step 1.1
Differentiate.
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 .
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 1.3
Evaluate .
Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 2
Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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 .
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 2.4
Differentiate using the Constant Rule.
Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
Add and .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Step 4.1
Find the first derivative.
Step 4.1.1
Differentiate.
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 .
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 4.1.3
Evaluate .
Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3.3
Multiply by .
Step 4.2
The first derivative of with respect to is .
Step 5
Step 5.1
Set the first derivative equal to .
Step 5.2
Factor by grouping.
Step 5.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 5.2.1.1
Factor out of .
Step 5.2.1.2
Rewrite as plus
Step 5.2.1.3
Apply the distributive property.
Step 5.2.2
Factor out the greatest common factor from each group.
Step 5.2.2.1
Group the first two terms and the last two terms.
Step 5.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.2.3
Factor the polynomial by factoring out the greatest common factor, .
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 and solve for .
Step 5.4.1
Set equal to .
Step 5.4.2
Solve for .
Step 5.4.2.1
Add to both sides of the equation.
Step 5.4.2.2
Divide each term in by and simplify.
Step 5.4.2.2.1
Divide each term in by .
Step 5.4.2.2.2
Simplify the left side.
Step 5.4.2.2.2.1
Cancel the common factor of .
Step 5.4.2.2.2.1.1
Cancel the common factor.
Step 5.4.2.2.2.1.2
Divide by .
Step 5.5
Set equal to and solve for .
Step 5.5.1
Set equal to .
Step 5.5.2
Add to both sides of the equation.
Step 5.6
The final solution is all the values that make true.
Step 6
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
Step 9.1
Simplify each term.
Step 9.1.1
Cancel the common factor of .
Step 9.1.1.1
Factor out of .
Step 9.1.1.2
Cancel the common factor.
Step 9.1.1.3
Rewrite the expression.
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
Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
Step 11.2.1
Simplify each term.
Step 11.2.1.1
Apply the product rule to .
Step 11.2.1.2
Raise to the power of .
Step 11.2.1.3
Raise to the power of .
Step 11.2.1.4
Apply the product rule to .
Step 11.2.1.5
Raise to the power of .
Step 11.2.1.6
Raise to the power of .
Step 11.2.1.7
Multiply .
Step 11.2.1.7.1
Combine and .
Step 11.2.1.7.2
Multiply by .
Step 11.2.1.8
Move the negative in front of the fraction.
Step 11.2.1.9
Multiply .
Step 11.2.1.9.1
Combine and .
Step 11.2.1.9.2
Multiply by .
Step 11.2.2
Find the common denominator.
Step 11.2.2.1
Multiply by .
Step 11.2.2.2
Multiply by .
Step 11.2.2.3
Multiply by .
Step 11.2.2.4
Multiply by .
Step 11.2.2.5
Reorder the factors of .
Step 11.2.2.6
Multiply by .
Step 11.2.2.7
Multiply by .
Step 11.2.3
Combine the numerators over the common denominator.
Step 11.2.4
Simplify each term.
Step 11.2.4.1
Multiply by .
Step 11.2.4.2
Multiply by .
Step 11.2.5
Simplify by adding and subtracting.
Step 11.2.5.1
Subtract from .
Step 11.2.5.2
Add and .
Step 11.2.6
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
Step 13.1
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
Step 15.1
Replace the variable with in the expression.
Step 15.2
Simplify the result.
Step 15.2.1
Simplify each term.
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 15.2.1.4
Multiply by .
Step 15.2.2
Simplify by adding and subtracting.
Step 15.2.2.1
Subtract from .
Step 15.2.2.2
Add and .
Step 15.2.3
The final answer is .
Step 16
These are the local extrema for .
is a local maxima
is a local minima
Step 17