# Calculus Examples

Step 1

Step 1.1

Find the first derivative.

Step 1.1.1

By the Sum Rule, the derivative of with respect to is .

Step 1.1.2

Evaluate .

Step 1.1.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.1.2.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2.3

Multiply by .

Step 1.1.3

Evaluate .

Step 1.1.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.1.3.2

Differentiate using the Power Rule which states that is where .

Step 1.1.3.3

Multiply by .

Step 1.2

Find the second derivative.

Step 1.2.1

By the Sum Rule, the derivative of with respect to is .

Step 1.2.2

Evaluate .

Step 1.2.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.2.2.2

Differentiate using the Power Rule which states that is where .

Step 1.2.2.3

Multiply by .

Step 1.2.3

Evaluate .

Step 1.2.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.2.3.2

Differentiate using the Power Rule which states that is where .

Step 1.2.3.3

Multiply by .

Step 1.3

The second derivative of with respect to is .

Step 2

Step 2.1

Set the second derivative equal to .

Step 2.2

Add to both sides of the equation.

Step 2.3

Divide each term in by and simplify.

Step 2.3.1

Divide each term in by .

Step 2.3.2

Simplify the left side.

Step 2.3.2.1

Cancel the common factor of .

Step 2.3.2.1.1

Cancel the common factor.

Step 2.3.2.1.2

Divide by .

Step 2.3.3

Simplify the right side.

Step 2.3.3.1

Cancel the common factor of and .

Step 2.3.3.1.1

Factor out of .

Step 2.3.3.1.2

Cancel the common factors.

Step 2.3.3.1.2.1

Factor out of .

Step 2.3.3.1.2.2

Cancel the common factor.

Step 2.3.3.1.2.3

Rewrite the expression.

Step 3

Step 3.1

Substitute in to find the value of .

Step 3.1.1

Replace the variable with in the expression.

Step 3.1.2

Simplify the result.

Step 3.1.2.1

Simplify each term.

Step 3.1.2.1.1

Apply the product rule to .

Step 3.1.2.1.2

One to any power is one.

Step 3.1.2.1.3

Raise to the power of .

Step 3.1.2.1.4

Combine and .

Step 3.1.2.1.5

Apply the product rule to .

Step 3.1.2.1.6

One to any power is one.

Step 3.1.2.1.7

Raise to the power of .

Step 3.1.2.1.8

Combine and .

Step 3.1.2.1.9

Move the negative in front of the fraction.

Step 3.1.2.2

To write as a fraction with a common denominator, multiply by .

Step 3.1.2.3

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Step 3.1.2.3.1

Multiply by .

Step 3.1.2.3.2

Multiply by .

Step 3.1.2.4

Combine the numerators over the common denominator.

Step 3.1.2.5

Simplify the numerator.

Step 3.1.2.5.1

Multiply by .

Step 3.1.2.5.2

Subtract from .

Step 3.1.2.6

Move the negative in front of the fraction.

Step 3.1.2.7

The final answer is .

Step 3.2

The point found by substituting in is . This point can be an inflection point.

Step 4

Split into intervals around the points that could potentially be inflection points.

Step 5

Step 5.1

Replace the variable with in the expression.

Step 5.2

Simplify the result.

Step 5.2.1

Multiply by .

Step 5.2.2

Subtract from .

Step 5.2.3

The final answer is .

Step 5.3

At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval

Decreasing on since

Decreasing on since

Step 6

Step 6.1

Replace the variable with in the expression.

Step 6.2

Simplify the result.

Step 6.2.1

Multiply by .

Step 6.2.2

Subtract from .

Step 6.2.3

The final answer is .

Step 6.3

At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .

Increasing on since

Increasing on since

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is .

Step 8