Find the second derivative.

Find the first derivative.

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

Differentiate using the Product Rule which states that is where and .

Differentiate using the Exponential Rule which states that is where =.

Differentiate using the Power Rule.

Differentiate using the Power Rule which states that is where .

Multiply by .

Simplify.

Apply the distributive property.

Reorder terms.

Find the second derivative.

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

Evaluate .

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

Differentiate using the Product Rule which states that is where and .

Differentiate using the Power Rule which states that is where .

Differentiate using the Exponential Rule which states that is where =.

Multiply by .

Evaluate .

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

Differentiate using the Exponential Rule which states that is where =.

Simplify.

Apply the distributive property.

Add and .

Reorder terms.

The second derivative of with respect to is .

Set the second derivative equal to then solve the equation .

Reorder factors in .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Set equal to and solve for .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

Use logarithm rules to move out of the exponent.

The natural logarithm of is .

Multiply by .

The equation cannot be solved because it is undefined.

Undefined

Since the logarithm is undefined, there is no solution.

No solution

No solution

Set equal to and solve for .

Set the factor equal to .

Subtract from both sides of the equation.

Find the points where the second derivative is .

Substitute in to find the value of .

Replace the variable with in the expression.

Simplify the result.

Multiply by .

Rewrite the expression using the negative exponent rule .

Combine and .

Move the negative in front of the fraction.

The final answer is .

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

Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Rewrite the expression using the negative exponent rule .

Combine and .

Multiply .

Combine and .

Multiply by .

Move the negative in front of the fraction.

Replace with an approximation.

Raise to the power of .

Divide by .

Multiply by .

Rewrite the expression using the negative exponent rule .

Combine and .

Add and .

The final answer is .

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

Decreasing on since

Decreasing on since

Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Rewrite the expression using the negative exponent rule .

Combine and .

Multiply .

Combine and .

Multiply by .

Move the negative in front of the fraction.

Replace with an approximation.

Raise to the power of .

Divide by .

Multiply by .

Rewrite the expression using the negative exponent rule .

Combine and .

Add and .

The final answer is .

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

Increasing on since

Increasing on since

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 .

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.

Interval Notation:

Set-Builder Notation:

Create intervals around the inflection points and the undefined values.

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Rewrite the expression using the negative exponent rule .

Combine and .

Multiply .

Combine and .

Multiply by .

Move the negative in front of the fraction.

Rewrite the expression using the negative exponent rule .

Combine and .

Simplify the expression.

Combine the numerators over the common denominator.

Add and .

Move the negative in front of the fraction.

The final answer is .

The graph is concave down on the interval because is negative.

Concave down on since is negative

Concave down on since is negative

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Anything raised to is .

Multiply by .

Multiply by .

Anything raised to is .

Multiply by .

Add and .

The final answer is .

The graph is concave up on the interval because is positive.

Concave up on since is positive

Concave up on since is positive

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on since is negative

Concave up on since is positive

Find the Concavity f(x)=2xe^x