Calculus Examples

Find the Concavity f(x)=1/x
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
Rewrite as .
Step 1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3
Rewrite the expression using the negative exponent rule .
Step 1.1.2
Find the second derivative.
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Step 1.1.2.1
Differentiate using the Product Rule which states that is where and .
Step 1.1.2.2
Differentiate.
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Step 1.1.2.2.1
Rewrite as .
Step 1.1.2.2.2
Multiply the exponents in .
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Step 1.1.2.2.2.1
Apply the power rule and multiply exponents, .
Step 1.1.2.2.2.2
Multiply by .
Step 1.1.2.2.3
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2.4
Multiply by .
Step 1.1.2.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2.6
Simplify the expression.
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Step 1.1.2.2.6.1
Multiply by .
Step 1.1.2.2.6.2
Add and .
Step 1.1.2.3
Simplify.
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Step 1.1.2.3.1
Rewrite the expression using the negative exponent rule .
Step 1.1.2.3.2
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 .
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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
Since , there are no solutions.
No solution
No solution
No solution
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
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
Cancel the common factor of and .
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Step 4.2.1.1
Rewrite as .
Step 4.2.1.2
Factor out of .
Step 4.2.1.3
Cancel the common factors.
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Step 4.2.1.3.1
Factor out of .
Step 4.2.1.3.2
Cancel the common factor.
Step 4.2.1.3.3
Rewrite the expression.
Step 4.2.2
Simplify the expression.
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Step 4.2.2.1
Raise to the power of .
Step 4.2.2.2
Move the negative in front of the fraction.
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.
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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
Cancel the common factor of and .
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Step 5.2.1.1
Factor out of .
Step 5.2.1.2
Cancel the common factors.
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Step 5.2.1.2.1
Factor out of .
Step 5.2.1.2.2
Cancel the common factor.
Step 5.2.1.2.3
Rewrite the expression.
Step 5.2.2
Raise to the power of .
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