Calculus Examples

Find the Concavity f(x)=x/(x^2-9)
Step 1
Find the values where the second derivative is equal to .
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Step 1.1
Find the second derivative.
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Step 1.1.1
Find the first derivative.
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Step 1.1.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.1.2
Differentiate.
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Step 1.1.1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.2
Multiply by .
Step 1.1.1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2.4
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.6
Simplify the expression.
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Step 1.1.1.2.6.1
Add and .
Step 1.1.1.2.6.2
Multiply by .
Step 1.1.1.3
Raise to the power of .
Step 1.1.1.4
Raise to the power of .
Step 1.1.1.5
Use the power rule to combine exponents.
Step 1.1.1.6
Add and .
Step 1.1.1.7
Subtract from .
Step 1.1.2
Find the second derivative.
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Step 1.1.2.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.2.2
Differentiate.
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Step 1.1.2.2.1
Multiply the exponents in .
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Step 1.1.2.2.1.1
Apply the power rule and multiply exponents, .
Step 1.1.2.2.1.2
Multiply by .
Step 1.1.2.2.2
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2.4
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2.5
Multiply by .
Step 1.1.2.2.6
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2.7
Add and .
Step 1.1.2.3
Differentiate using the chain rule, which states that is where and .
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Step 1.1.2.3.1
To apply the Chain Rule, set as .
Step 1.1.2.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3.3
Replace all occurrences of with .
Step 1.1.2.4
Differentiate.
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Step 1.1.2.4.1
Multiply by .
Step 1.1.2.4.2
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.4.3
Differentiate using the Power Rule which states that is where .
Step 1.1.2.4.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.4.5
Simplify the expression.
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Step 1.1.2.4.5.1
Add and .
Step 1.1.2.4.5.2
Move to the left of .
Step 1.1.2.4.5.3
Multiply by .
Step 1.1.2.5
Simplify.
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Step 1.1.2.5.1
Apply the distributive property.
Step 1.1.2.5.2
Apply the distributive property.
Step 1.1.2.5.3
Simplify the numerator.
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Step 1.1.2.5.3.1
Simplify each term.
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Step 1.1.2.5.3.1.1
Rewrite using the commutative property of multiplication.
Step 1.1.2.5.3.1.2
Rewrite as .
Step 1.1.2.5.3.1.3
Expand using the FOIL Method.
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Step 1.1.2.5.3.1.3.1
Apply the distributive property.
Step 1.1.2.5.3.1.3.2
Apply the distributive property.
Step 1.1.2.5.3.1.3.3
Apply the distributive property.
Step 1.1.2.5.3.1.4
Simplify and combine like terms.
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Step 1.1.2.5.3.1.4.1
Simplify each term.
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Step 1.1.2.5.3.1.4.1.1
Multiply by by adding the exponents.
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Step 1.1.2.5.3.1.4.1.1.1
Use the power rule to combine exponents.
Step 1.1.2.5.3.1.4.1.1.2
Add and .
Step 1.1.2.5.3.1.4.1.2
Move to the left of .
Step 1.1.2.5.3.1.4.1.3
Multiply by .
Step 1.1.2.5.3.1.4.2
Subtract from .
Step 1.1.2.5.3.1.5
Apply the distributive property.
Step 1.1.2.5.3.1.6
Simplify.
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Step 1.1.2.5.3.1.6.1
Multiply by .
Step 1.1.2.5.3.1.6.2
Multiply by .
Step 1.1.2.5.3.1.7
Apply the distributive property.
Step 1.1.2.5.3.1.8
Simplify.
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Step 1.1.2.5.3.1.8.1
Multiply by by adding the exponents.
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Step 1.1.2.5.3.1.8.1.1
Move .
Step 1.1.2.5.3.1.8.1.2
Multiply by .
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Step 1.1.2.5.3.1.8.1.2.1
Raise to the power of .
Step 1.1.2.5.3.1.8.1.2.2
Use the power rule to combine exponents.
Step 1.1.2.5.3.1.8.1.3
Add and .
Step 1.1.2.5.3.1.8.2
Multiply by by adding the exponents.
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Step 1.1.2.5.3.1.8.2.1
Move .
Step 1.1.2.5.3.1.8.2.2
Multiply by .
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Step 1.1.2.5.3.1.8.2.2.1
Raise to the power of .
Step 1.1.2.5.3.1.8.2.2.2
Use the power rule to combine exponents.
Step 1.1.2.5.3.1.8.2.3
Add and .
Step 1.1.2.5.3.1.9
Simplify each term.
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Step 1.1.2.5.3.1.9.1
Multiply by .
Step 1.1.2.5.3.1.9.2
Multiply by .
Step 1.1.2.5.3.1.10
Multiply by by adding the exponents.
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Step 1.1.2.5.3.1.10.1
Multiply by .
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Step 1.1.2.5.3.1.10.1.1
Raise to the power of .
Step 1.1.2.5.3.1.10.1.2
Use the power rule to combine exponents.
Step 1.1.2.5.3.1.10.2
Add and .
Step 1.1.2.5.3.1.11
Expand using the FOIL Method.
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Step 1.1.2.5.3.1.11.1
Apply the distributive property.
Step 1.1.2.5.3.1.11.2
Apply the distributive property.
Step 1.1.2.5.3.1.11.3
Apply the distributive property.
Step 1.1.2.5.3.1.12
Simplify and combine like terms.
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Step 1.1.2.5.3.1.12.1
Simplify each term.
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Step 1.1.2.5.3.1.12.1.1
Multiply by by adding the exponents.
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Step 1.1.2.5.3.1.12.1.1.1
Move .
Step 1.1.2.5.3.1.12.1.1.2
Use the power rule to combine exponents.
Step 1.1.2.5.3.1.12.1.1.3
Add and .
Step 1.1.2.5.3.1.12.1.2
Rewrite using the commutative property of multiplication.
Step 1.1.2.5.3.1.12.1.3
Multiply by by adding the exponents.
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Step 1.1.2.5.3.1.12.1.3.1
Move .
Step 1.1.2.5.3.1.12.1.3.2
Multiply by .
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Step 1.1.2.5.3.1.12.1.3.2.1
Raise to the power of .
Step 1.1.2.5.3.1.12.1.3.2.2
Use the power rule to combine exponents.
Step 1.1.2.5.3.1.12.1.3.3
Add and .
Step 1.1.2.5.3.1.12.1.4
Multiply by .
Step 1.1.2.5.3.1.12.1.5
Multiply by .
Step 1.1.2.5.3.1.12.2
Add and .
Step 1.1.2.5.3.1.12.3
Add and .
Step 1.1.2.5.3.2
Add and .
Step 1.1.2.5.3.3
Subtract from .
Step 1.1.2.5.4
Simplify the numerator.
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Step 1.1.2.5.4.1
Factor out of .
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Step 1.1.2.5.4.1.1
Factor out of .
Step 1.1.2.5.4.1.2
Factor out of .
Step 1.1.2.5.4.1.3
Factor out of .
Step 1.1.2.5.4.1.4
Factor out of .
Step 1.1.2.5.4.1.5
Factor out of .
Step 1.1.2.5.4.2
Rewrite as .
Step 1.1.2.5.4.3
Let . Substitute for all occurrences of .
Step 1.1.2.5.4.4
Factor using the AC method.
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Step 1.1.2.5.4.4.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 1.1.2.5.4.4.2
Write the factored form using these integers.
Step 1.1.2.5.4.5
Replace all occurrences of with .
Step 1.1.2.5.4.6
Rewrite as .
Step 1.1.2.5.4.7
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.1.2.5.5
Simplify the denominator.
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Step 1.1.2.5.5.1
Rewrite as .
Step 1.1.2.5.5.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.1.2.5.5.3
Apply the product rule to .
Step 1.1.2.5.6
Cancel the common factor of and .
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Step 1.1.2.5.6.1
Factor out of .
Step 1.1.2.5.6.2
Cancel the common factors.
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Step 1.1.2.5.6.2.1
Factor out of .
Step 1.1.2.5.6.2.2
Cancel the common factor.
Step 1.1.2.5.6.2.3
Rewrite the expression.
Step 1.1.2.5.7
Cancel the common factor of and .
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Step 1.1.2.5.7.1
Factor out of .
Step 1.1.2.5.7.2
Cancel the common factors.
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Step 1.1.2.5.7.2.1
Factor out of .
Step 1.1.2.5.7.2.2
Cancel the common factor.
Step 1.1.2.5.7.2.3
Rewrite the expression.
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 .
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Step 1.2.1
Set the second derivative equal to .
Step 1.2.2
Set the numerator equal to zero.
Step 1.2.3
Solve the equation for .
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Step 1.2.3.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.3.2
Set equal to .
Step 1.2.3.3
Set equal to and solve for .
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Step 1.2.3.3.1
Set equal to .
Step 1.2.3.3.2
Solve for .
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Step 1.2.3.3.2.1
Subtract from both sides of the equation.
Step 1.2.3.3.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.3.3.2.3
Simplify .
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Step 1.2.3.3.2.3.1
Rewrite as .
Step 1.2.3.3.2.3.2
Rewrite as .
Step 1.2.3.3.2.3.3
Rewrite as .
Step 1.2.3.3.2.3.4
Rewrite as .
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Step 1.2.3.3.2.3.4.1
Factor out of .
Step 1.2.3.3.2.3.4.2
Rewrite as .
Step 1.2.3.3.2.3.5
Pull terms out from under the radical.
Step 1.2.3.3.2.3.6
Move to the left of .
Step 1.2.3.3.2.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.3.3.2.4.1
First, use the positive value of the to find the first solution.
Step 1.2.3.3.2.4.2
Next, use the negative value of the to find the second solution.
Step 1.2.3.3.2.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.2.3.4
The final solution is all the values that make true.
Step 2
Find the domain of .
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Step 2.1
Set the denominator in equal to to find where the expression is undefined.
Step 2.2
Solve for .
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Step 2.2.1
Add to both sides of the equation.
Step 2.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.2.3
Simplify .
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Step 2.2.3.1
Rewrite as .
Step 2.2.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 2.2.4.1
First, use the positive value of the to find the first solution.
Step 2.2.4.2
Next, use the negative value of the to find the second solution.
Step 2.2.4.3
The complete solution is the result of both the positive and negative portions of the solution.
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.
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Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
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Step 4.2.1
Multiply by .
Step 4.2.2
Simplify the denominator.
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Step 4.2.2.1
Add and .
Step 4.2.2.2
Subtract from .
Step 4.2.2.3
Raise to the power of .
Step 4.2.2.4
Raise to the power of .
Step 4.2.3
Simplify the numerator.
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Step 4.2.3.1
Raise to the power of .
Step 4.2.3.2
Add and .
Step 4.2.4
Reduce the expression by cancelling the common factors.
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Step 4.2.4.1
Multiply by .
Step 4.2.4.2
Multiply by .
Step 4.2.4.3
Cancel the common factor of and .
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Step 4.2.4.3.1
Factor out of .
Step 4.2.4.3.2
Cancel the common factors.
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Step 4.2.4.3.2.1
Factor out of .
Step 4.2.4.3.2.2
Cancel the common factor.
Step 4.2.4.3.2.3
Rewrite the expression.
Step 4.2.4.4
Move the negative in front of the fraction.
Step 4.2.5
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.
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Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
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Step 5.2.1
Multiply by .
Step 5.2.2
Simplify the denominator.
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Step 5.2.2.1
Add and .
Step 5.2.2.2
Subtract from .
Step 5.2.2.3
One to any power is one.
Step 5.2.2.4
Raise to the power of .
Step 5.2.2.5
Multiply by .
Step 5.2.3
Simplify the numerator.
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Step 5.2.3.1
Raise to the power of .
Step 5.2.3.2
Add and .
Step 5.2.4
Reduce the expression by cancelling the common factors.
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Step 5.2.4.1
Multiply by .
Step 5.2.4.2
Dividing two negative values results in a positive value.
Step 5.2.5
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
Substitute any number from the interval into the second derivative and evaluate to determine the concavity.
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Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
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Step 6.2.1
Multiply by .
Step 6.2.2
Simplify the denominator.
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Step 6.2.2.1
Add and .
Step 6.2.2.2
Subtract from .
Step 6.2.2.3
Raise to the power of .
Step 6.2.2.4
Raise to the power of .
Step 6.2.3
Simplify the numerator.
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Step 6.2.3.1
Raise to the power of .
Step 6.2.3.2
Add and .
Step 6.2.4
Simplify the expression.
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Step 6.2.4.1
Multiply by .
Step 6.2.4.2
Multiply by .
Step 6.2.4.3
Move the negative in front of the fraction.
Step 6.2.5
The final answer is .
Step 6.3
The graph is concave down on the interval because is negative.
Concave down on since is negative
Concave down on since is negative
Step 7
Substitute any number from the interval into the second derivative and evaluate to determine the concavity.
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Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
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Step 7.2.1
Multiply by .
Step 7.2.2
Simplify the denominator.
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Step 7.2.2.1
Add and .
Step 7.2.2.2
Subtract from .
Step 7.2.2.3
Raise to the power of .
Step 7.2.2.4
Raise to the power of .
Step 7.2.3
Simplify the numerator.
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Step 7.2.3.1
Raise to the power of .
Step 7.2.3.2
Add and .
Step 7.2.4
Reduce the expression by cancelling the common factors.
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Step 7.2.4.1
Multiply by .
Step 7.2.4.2
Multiply by .
Step 7.2.4.3
Cancel the common factor of and .
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Step 7.2.4.3.1
Factor out of .
Step 7.2.4.3.2
Cancel the common factors.
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Step 7.2.4.3.2.1
Factor out of .
Step 7.2.4.3.2.2
Cancel the common factor.
Step 7.2.4.3.2.3
Rewrite the expression.
Step 7.2.5
The final answer is .
Step 7.3
The graph is concave up on the interval because is positive.
Concave up on since is positive
Concave up on since is positive
Step 8
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
Concave down on since is negative
Concave up on since is positive
Step 9