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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
Differentiate.
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 .
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
Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
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 .
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 3.4
Differentiate using the Constant Rule.
Step 3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.2
Add and .
Step 4
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 5
Step 5.1
Find the first derivative.
Step 5.1.1
Differentiate.
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 .
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 5.1.3
Evaluate .
Step 5.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3.2
Differentiate using the Power Rule which states that is where .
Step 5.1.3.3
Multiply by .
Step 5.1.4
Differentiate using the Constant Rule.
Step 5.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.4.2
Add and .
Step 5.2
The first derivative of with respect to is .
Step 6
Step 6.1
Set the first derivative equal to .
Step 6.2
Factor by grouping.
Step 6.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 6.2.1.1
Factor out of .
Step 6.2.1.2
Rewrite as plus
Step 6.2.1.3
Apply the distributive property.
Step 6.2.2
Factor out the greatest common factor from each group.
Step 6.2.2.1
Group the first two terms and the last two terms.
Step 6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 6.2.3
Factor the polynomial by factoring out the greatest common factor, .
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 and solve for .
Step 6.4.1
Set equal to .
Step 6.4.2
Solve for .
Step 6.4.2.1
Subtract from both sides of the equation.
Step 6.4.2.2
Divide each term in by and simplify.
Step 6.4.2.2.1
Divide each term in by .
Step 6.4.2.2.2
Simplify the left side.
Step 6.4.2.2.2.1
Cancel the common factor of .
Step 6.4.2.2.2.1.1
Cancel the common factor.
Step 6.4.2.2.2.1.2
Divide by .
Step 6.4.2.2.3
Simplify the right side.
Step 6.4.2.2.3.1
Move the negative in front of the fraction.
Step 6.5
Set equal to and solve for .
Step 6.5.1
Set equal to .
Step 6.5.2
Add to both sides of the equation.
Step 6.6
The final solution is all the values that make true.
Step 7
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
Step 10.1
Simplify each term.
Step 10.1.1
Cancel the common factor of .
Step 10.1.1.1
Move the leading negative in into the numerator.
Step 10.1.1.2
Factor out of .
Step 10.1.1.3
Cancel the common factor.
Step 10.1.1.4
Rewrite the expression.
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
Step 12.1
Replace the variable with in the expression.
Step 12.2
Simplify the result.
Step 12.2.1
Simplify each term.
Step 12.2.1.1
Use the power rule to distribute the exponent.
Step 12.2.1.1.1
Apply the product rule to .
Step 12.2.1.1.2
Apply the product rule to .
Step 12.2.1.2
Raise to the power of .
Step 12.2.1.3
Raise to the power of .
Step 12.2.1.4
Raise to the power of .
Step 12.2.1.5
Use the power rule to distribute the exponent.
Step 12.2.1.5.1
Apply the product rule to .
Step 12.2.1.5.2
Apply the product rule to .
Step 12.2.1.6
Raise to the power of .
Step 12.2.1.7
Multiply by .
Step 12.2.1.8
Raise to the power of .
Step 12.2.1.9
Raise to the power of .
Step 12.2.1.10
Multiply .
Step 12.2.1.10.1
Combine and .
Step 12.2.1.10.2
Multiply by .
Step 12.2.1.11
Move the negative in front of the fraction.
Step 12.2.1.12
Multiply .
Step 12.2.1.12.1
Multiply by .
Step 12.2.1.12.2
Combine and .
Step 12.2.1.12.3
Multiply by .
Step 12.2.2
Find the common denominator.
Step 12.2.2.1
Multiply by .
Step 12.2.2.2
Multiply by .
Step 12.2.2.3
Multiply by .
Step 12.2.2.4
Multiply by .
Step 12.2.2.5
Write as a fraction with denominator .
Step 12.2.2.6
Multiply by .
Step 12.2.2.7
Multiply by .
Step 12.2.2.8
Reorder the factors of .
Step 12.2.2.9
Multiply by .
Step 12.2.2.10
Multiply by .
Step 12.2.3
Combine the numerators over the common denominator.
Step 12.2.4
Simplify each term.
Step 12.2.4.1
Multiply by .
Step 12.2.4.2
Multiply by .
Step 12.2.4.3
Multiply by .
Step 12.2.5
Simplify by adding and subtracting.
Step 12.2.5.1
Subtract from .
Step 12.2.5.2
Add and .
Step 12.2.5.3
Add and .
Step 12.2.6
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
Step 14.1
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
Step 16.1
Replace the variable with in the expression.
Step 16.2
Simplify the result.
Step 16.2.1
Simplify each term.
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 16.2.1.4
Multiply by .
Step 16.2.2
Simplify by adding and subtracting.
Step 16.2.2.1
Subtract from .
Step 16.2.2.2
Subtract from .
Step 16.2.2.3
Add and .
Step 16.2.3
The final answer is .
Step 17
These are the local extrema for .
is a local maxima
is a local minima
Step 18