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Calculus Examples
Step 1
Step 1.1
By the Sum Rule, the derivative of with respect to is .
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 1.4
Evaluate .
Step 1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.4.3
Multiply by .
Step 1.5
Differentiate using the Constant Rule.
Step 1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.5.2
Add and .
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
By the Sum Rule, the derivative of with respect to is .
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 4.1.4
Evaluate .
Step 4.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4.2
Differentiate using the Power Rule which states that is where .
Step 4.1.4.3
Multiply by .
Step 4.1.5
Differentiate using the Constant Rule.
Step 4.1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.5.2
Add and .
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
Substitute into the equation. This will make the quadratic formula easy to use.
Step 5.3
Use the quadratic formula to find the solutions.
Step 5.4
Substitute the values , , and into the quadratic formula and solve for .
Step 5.5
Simplify.
Step 5.5.1
Simplify the numerator.
Step 5.5.1.1
Raise to the power of .
Step 5.5.1.2
Multiply .
Step 5.5.1.2.1
Multiply by .
Step 5.5.1.2.2
Multiply by .
Step 5.5.1.3
Subtract from .
Step 5.5.1.4
Rewrite as .
Step 5.5.1.4.1
Factor out of .
Step 5.5.1.4.2
Rewrite as .
Step 5.5.1.5
Pull terms out from under the radical.
Step 5.5.2
Multiply by .
Step 5.5.3
Simplify .
Step 5.6
Simplify the expression to solve for the portion of the .
Step 5.6.1
Simplify the numerator.
Step 5.6.1.1
Raise to the power of .
Step 5.6.1.2
Multiply .
Step 5.6.1.2.1
Multiply by .
Step 5.6.1.2.2
Multiply by .
Step 5.6.1.3
Subtract from .
Step 5.6.1.4
Rewrite as .
Step 5.6.1.4.1
Factor out of .
Step 5.6.1.4.2
Rewrite as .
Step 5.6.1.5
Pull terms out from under the radical.
Step 5.6.2
Multiply by .
Step 5.6.3
Simplify .
Step 5.6.4
Change the to .
Step 5.7
Simplify the expression to solve for the portion of the .
Step 5.7.1
Simplify the numerator.
Step 5.7.1.1
Raise to the power of .
Step 5.7.1.2
Multiply .
Step 5.7.1.2.1
Multiply by .
Step 5.7.1.2.2
Multiply by .
Step 5.7.1.3
Subtract from .
Step 5.7.1.4
Rewrite as .
Step 5.7.1.4.1
Factor out of .
Step 5.7.1.4.2
Rewrite as .
Step 5.7.1.5
Pull terms out from under the radical.
Step 5.7.2
Multiply by .
Step 5.7.3
Simplify .
Step 5.7.4
Change the to .
Step 5.8
The final answer is the combination of both solutions.
Step 5.9
Substitute the real value of back into the solved equation.
Step 5.10
Solve the first equation for .
Step 5.11
Solve the equation for .
Step 5.11.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.11.2
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.11.2.1
First, use the positive value of the to find the first solution.
Step 5.11.2.2
Next, use the negative value of the to find the second solution.
Step 5.11.2.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.12
Solve the second equation for .
Step 5.13
Solve the equation for .
Step 5.13.1
Remove parentheses.
Step 5.13.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.13.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.13.3.1
First, use the positive value of the to find the first solution.
Step 5.13.3.2
Next, use the negative value of the to find the second solution.
Step 5.13.3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.14
The solution to is .
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
Rewrite as .
Step 9.2
Raise to the power of .
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
Rewrite as .
Step 11.2.1.2
Raise to the power of .
Step 11.2.1.3
Rewrite as .
Step 11.2.1.4
Raise to the power of .
Step 11.2.2
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
Apply the product rule to .
Step 13.2
Raise to the power of .
Step 13.3
Rewrite as .
Step 13.4
Raise to the power of .
Step 13.5
Multiply by .
Step 13.6
Multiply by .
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
Apply the product rule to .
Step 15.2.1.2
Multiply by by adding the exponents.
Step 15.2.1.2.1
Move .
Step 15.2.1.2.2
Multiply by .
Step 15.2.1.2.2.1
Raise to the power of .
Step 15.2.1.2.2.2
Use the power rule to combine exponents.
Step 15.2.1.2.3
Add and .
Step 15.2.1.3
Raise to the power of .
Step 15.2.1.4
Multiply by .
Step 15.2.1.5
Rewrite as .
Step 15.2.1.6
Raise to the power of .
Step 15.2.1.7
Apply the product rule to .
Step 15.2.1.8
Raise to the power of .
Step 15.2.1.9
Rewrite as .
Step 15.2.1.10
Raise to the power of .
Step 15.2.1.11
Multiply by .
Step 15.2.1.12
Multiply by .
Step 15.2.2
The final answer is .
Step 16
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 17
Step 17.1
Rewrite as .
Step 17.2
Raise to the power of .
Step 18
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 19
Step 19.1
Replace the variable with in the expression.
Step 19.2
Simplify the result.
Step 19.2.1
Simplify each term.
Step 19.2.1.1
Rewrite as .
Step 19.2.1.2
Raise to the power of .
Step 19.2.1.3
Rewrite as .
Step 19.2.1.4
Raise to the power of .
Step 19.2.2
The final answer is .
Step 20
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 21
Step 21.1
Apply the product rule to .
Step 21.2
Raise to the power of .
Step 21.3
Rewrite as .
Step 21.4
Raise to the power of .
Step 21.5
Multiply by .
Step 21.6
Multiply by .
Step 22
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 23
Step 23.1
Replace the variable with in the expression.
Step 23.2
Simplify the result.
Step 23.2.1
Simplify each term.
Step 23.2.1.1
Apply the product rule to .
Step 23.2.1.2
Multiply by by adding the exponents.
Step 23.2.1.2.1
Move .
Step 23.2.1.2.2
Multiply by .
Step 23.2.1.2.2.1
Raise to the power of .
Step 23.2.1.2.2.2
Use the power rule to combine exponents.
Step 23.2.1.2.3
Add and .
Step 23.2.1.3
Raise to the power of .
Step 23.2.1.4
Multiply by .
Step 23.2.1.5
Rewrite as .
Step 23.2.1.6
Raise to the power of .
Step 23.2.1.7
Apply the product rule to .
Step 23.2.1.8
Raise to the power of .
Step 23.2.1.9
Rewrite as .
Step 23.2.1.10
Raise to the power of .
Step 23.2.1.11
Multiply by .
Step 23.2.1.12
Multiply by .
Step 23.2.2
The final answer is .
Step 24
These are the local extrema for .
is a local maxima
is a local minima
is a local minima
is a local maxima
Step 25