Calculus Examples

Step 1
Find the values where the second derivative is equal to .
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Find the second derivative.
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Find the first derivative.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Add and .
Find the second derivative.
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
The second derivative of with respect to is .
Set the second derivative equal to then solve the equation .
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Set the second derivative equal to .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Divide by .
Take the square root of both sides of the equation to eliminate the exponent on the left side.
Simplify .
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Plus or minus is .
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.
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Replace the variable with in the expression.
Simplify the result.
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Raise to the power of .
Multiply by .
The final answer is .
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.
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Replace the variable with in the expression.
Simplify the result.
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Raise to the power of .
Multiply by .
The final answer is .
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
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