James F. answered • 06/20/14

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Statistics Graduate Student and Tutor

To get the concavity, we need to take the second derivative of the function.

First, take f'(x) = 16/(x+2)^2 using the quotient rule

Next, take f''(x) = -32/(x+2)^3 using the quotient rule on the first derivative

Solving f''(x) = 0 would normally give the point of inflection, but in this case we'll get no solution, so we will inspect the critical value of x = -2.

Next, check a value smaller than x = -2 and a value larger than x = -2 to test for concavity (negative implies concave down).

Let me know if this gets you going in the right direction!

James F.

What do you think the answers are? I'd be happy to validate them.

J.T.

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06/20/14

Jessica S.

if i plug in lets say -3 and -1 in the second deritative function, i get solutions such as -32 however how would i denote that in interval notation?

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06/20/14

James F.

So the second derivative is -32/(x+2)^3 <---I had a typo above before

We plug in x = -3 and get 32

We plug in x = -1 and get -32

Since x = -2 is a critical value, we'll use it as the end point of our intervals.

f(x) is concave up on (-∞,-2) and concave down on (-2,∞). Now you can double check you intervals: if you pick ANY number less than -2, f''(x) will always be positive. If you pick any number greater than -2, f''(x) will always be negative.

J.T.

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06/21/14

Jessica S.

06/20/14