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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
Differentiate using the Constant Rule.
Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
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
Differentiate using the Constant Rule.
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
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
Differentiate using the Constant Rule.
Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
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
Add to both sides of the equation.
Step 5.3
Divide each term in by and simplify.
Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
Step 5.3.2.1
Cancel the common factor of .
Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Divide by .
Step 5.3.3
Simplify the right side.
Step 5.3.3.1
Divide by .
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
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 10
Step 10.1
Replace the variable with in the expression.
Step 10.2
Simplify the result.
Step 10.2.1
Simplify each term.
Step 10.2.1.1
Raise to the power of .
Step 10.2.1.2
Multiply by .
Step 10.2.2
Simplify by adding and subtracting.
Step 10.2.2.1
Subtract from .
Step 10.2.2.2
Add and .
Step 10.2.3
The final answer is .
Step 11
These are the local extrema for .
is a local minima
Step 12