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Calculus Examples
Step 1
Step 1.1
Find the second derivative.
Step 1.1.1
Find the first derivative.
Step 1.1.1.1
Differentiate using the chain rule, which states that is where and .
Step 1.1.1.1.1
To apply the Chain Rule, set as .
Step 1.1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.1.3
Replace all occurrences of with .
Step 1.1.1.2
Differentiate.
Step 1.1.1.2.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.4
Simplify the expression.
Step 1.1.1.2.4.1
Add and .
Step 1.1.1.2.4.2
Multiply by .
Step 1.1.2
Find the second derivative.
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 chain rule, which states that is where and .
Step 1.1.2.2.1
To apply the Chain Rule, set as .
Step 1.1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2.3
Replace all occurrences of with .
Step 1.1.2.3
Differentiate.
Step 1.1.2.3.1
Multiply by .
Step 1.1.2.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.3.3
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.3.5
Simplify the expression.
Step 1.1.2.3.5.1
Add and .
Step 1.1.2.3.5.2
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
Divide each term in by and simplify.
Step 1.2.2.1
Divide each term in by .
Step 1.2.2.2
Simplify the left side.
Step 1.2.2.2.1
Cancel the common factor of .
Step 1.2.2.2.1.1
Cancel the common factor.
Step 1.2.2.2.1.2
Divide by .
Step 1.2.2.3
Simplify the right side.
Step 1.2.2.3.1
Divide by .
Step 1.2.3
Set the equal to .
Step 1.2.4
Add to both sides of the equation.
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
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Subtract from .
Step 4.2.2
Raise to the power of .
Step 4.2.3
Multiply by .
Step 4.2.4
The final answer is .
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
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Subtract from .
Step 5.2.2
Raise to the power of .
Step 5.2.3
Multiply by .
Step 5.2.4
The final answer is .
Step 5.3
The graph is concave up on the interval because is positive.
Concave up on since is positive
Concave up on since is positive
Step 6
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 up on since is positive
Step 7