# Calculus Examples

Step 1
Find the values where the second derivative is equal to .
Step 1.1
Find the second derivative.
Step 1.1.1
Find the first derivative.
Step 1.1.1.1
Differentiate.
Step 1.1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2
Evaluate .
Step 1.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Multiply by .
Step 1.1.2
Find the second derivative.
Step 1.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.2
Evaluate .
Step 1.1.2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2.3
Multiply by .
Step 1.1.2.3
Evaluate .
Step 1.1.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3.3
Multiply by .
Step 1.1.3
The second derivative of with respect to is .
Step 1.2
Set the second derivative equal to then solve the equation .
Step 1.2.1
Set the second derivative equal to .
Step 1.2.2
Factor out of .
Step 1.2.2.1
Factor out of .
Step 1.2.2.2
Factor out of .
Step 1.2.2.3
Factor out of .
Step 1.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4
Set equal to .
Step 1.2.5
Set equal to and solve for .
Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Add to both sides of the equation.
Step 1.2.6
The final solution is all the values that make true.
Step 2
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:
Step 3
Create intervals around the -values where the second derivative is zero or undefined.
Step 4
Substitute any number from the interval into the second derivative and evaluate to determine the concavity.
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Raise to the power of .
Step 4.2.1.2
Multiply by .
Step 4.2.1.3
Multiply by .
Step 4.2.2
Step 4.2.3
Step 4.3
The graph is concave up on the interval because is positive.
Concave up on since is positive
Concave up on since is positive
Step 5
Substitute any number from the interval into the second derivative and evaluate to determine the concavity.
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
One to any power is one.
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Multiply by .
Step 5.2.2
Subtract from .
Step 5.2.3
Step 5.3
The graph is concave down on the interval because is negative.
Concave down on since is negative
Concave down on since is negative
Step 6
Substitute any number from the interval into the second derivative and evaluate to determine the concavity.
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
Simplify each term.
Step 6.2.1.1
Raise to the power of .
Step 6.2.1.2
Multiply by .
Step 6.2.1.3
Multiply by .
Step 6.2.2
Subtract from .
Step 6.2.3
Step 6.3
The graph is concave up on the interval because is positive.
Concave up on since is positive
Concave up on since is positive
Step 7
The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.
Concave up on since is positive
Concave down on since is negative
Concave up on since is positive
Step 8