Calculus Examples

Find the Concavity f(x)=(x+2)/(x-2)
Step 1
Find the values where the second derivative is equal to .
Tap for more steps...
Step 1.1
Find the second derivative.
Tap for more steps...
Step 1.1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.1.2
Differentiate.
Tap for more steps...
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.
Tap for more steps...
Step 1.1.1.2.4.1
Add and .
Step 1.1.1.2.4.2
Multiply by .
Step 1.1.1.2.5
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2.6
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.7
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.8
Simplify the expression.
Tap for more steps...
Step 1.1.1.2.8.1
Add and .
Step 1.1.1.2.8.2
Multiply by .
Step 1.1.1.3
Simplify.
Tap for more steps...
Step 1.1.1.3.1
Apply the distributive property.
Step 1.1.1.3.2
Simplify the numerator.
Tap for more steps...
Step 1.1.1.3.2.1
Combine the opposite terms in .
Tap for more steps...
Step 1.1.1.3.2.1.1
Subtract from .
Step 1.1.1.3.2.1.2
Subtract from .
Step 1.1.1.3.2.2
Multiply by .
Step 1.1.1.3.2.3
Subtract from .
Step 1.1.1.3.3
Move the negative in front of the fraction.
Step 1.1.2
Find the second derivative.
Tap for more steps...
Step 1.1.2.1
Differentiate using the Constant Multiple Rule.
Tap for more steps...
Step 1.1.2.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.1.2
Apply basic rules of exponents.
Tap for more steps...
Step 1.1.2.1.2.1
Rewrite as .
Step 1.1.2.1.2.2
Multiply the exponents in .
Tap for more steps...
Step 1.1.2.1.2.2.1
Apply the power rule and multiply exponents, .
Step 1.1.2.1.2.2.2
Multiply by .
Step 1.1.2.2
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
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.
Tap for more steps...
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.
Tap for more steps...
Step 1.1.2.3.5.1
Add and .
Step 1.1.2.3.5.2
Multiply by .
Step 1.1.2.4
Simplify.
Tap for more steps...
Step 1.1.2.4.1
Rewrite the expression using the negative exponent rule .
Step 1.1.2.4.2
Combine and .
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 .
Tap for more steps...
Step 1.2.1
Set the second derivative equal to .
Step 1.2.2
Set the numerator equal to zero.
Step 1.2.3
Since , there are no solutions.
No solution
No solution
No solution
Step 2
Find the domain of .
Tap for more steps...
Step 2.1
Set the denominator in equal to to find where the expression is undefined.
Step 2.2
Add to both sides of the equation.
Step 2.3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
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.
Tap for more steps...
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Tap for more steps...
Step 4.2.1
Simplify the denominator.
Tap for more steps...
Step 4.2.1.1
Subtract from .
Step 4.2.1.2
Raise to the power of .
Step 4.2.2
Divide by .
Step 4.2.3
The final answer is .
Step 4.3
The graph is concave down on the interval because is negative.
Concave down on since is negative
Concave down on since is negative
Step 5
Substitute any number from the interval into the second derivative and evaluate to determine the concavity.
Tap for more steps...
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Tap for more steps...
Step 5.2.1
Simplify the denominator.
Tap for more steps...
Step 5.2.1.1
Subtract from .
Step 5.2.1.2
Raise to the power of .
Step 5.2.2
Divide by .
Step 5.2.3
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 down on since is negative
Concave up on since is positive
Step 7