# 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

Differentiate using the Power Rule which states that is where .

Step 1.1.3

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

Step 1.1.4

Add and .

Step 1.2

Find the second derivative.

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

The second derivative of with respect to is .

Step 2

Step 2.1

Set the second derivative equal to .

Step 2.2

Divide each term in by and simplify.

Step 2.2.1

Divide each term in by .

Step 2.2.2

Simplify the left side.

Step 2.2.2.1

Cancel the common factor of .

Step 2.2.2.1.1

Cancel the common factor.

Step 2.2.2.1.2

Divide by .

Step 2.2.3

Simplify the right side.

Step 2.2.3.1

Divide by .

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

Step 2.4

Simplify .

Step 2.4.1

Rewrite as .

Step 2.4.2

Pull terms out from under the radical, assuming real numbers.

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

Raising to any positive power yields .

Step 3.1.2.2

Subtract from .

Step 3.1.2.3

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

Raise to the power of .

Step 5.2.2

Multiply by .

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

Raise to the power of .

Step 6.2.2

Multiply by .

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