Determine the intervals on which the following function is concave up or concave

Question

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f(x)= - e -x2/32 Determine the intervals on which the function is concave up or concave down. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. The function is concave up on and concave down on . (Type your answers in interval notation. Use a comma to separate answers as needed.) The function is concave up on . The function is never concave down. (Type your answer in interval notation.) The function is concave down on . The function is never concave up. (Type your answer in interval notation.) The function is never concave up nor concave down. Identify any inflection points. Select the correct choice below and, if necessary, fill in the answer box within your choice. There are inflection points at x = . (Use a comma to separate answers as needed.) There are no inflection points.

Explanation / Answer

f(x) = - e^(-x^2/ 32)

f ' (x ) =  - e^(-x^2/ 32) * -x/16 = 1/16 x e^(-x^2 /32)

next derivative is product rule:

f ''(x) = 1/16 x * (- e^(-x^2/ 32) * -x/16) + (- e^(-x^2/ 32) ) * 1/16  

factor out what is in common:

f''(x) = -1/16 e^(-x^2/ 32) ( x^2 / 16 - 1)

so basically (x^2 / 16 - 1) = 0 b/c the rest does not.

x^2 - 16 = 0

(x-4)(x+4) = 0

x = 4 and -4

set up a number line

- + -

---------------------------- -------

-4 4

So concave down (-inf, -4) U (4, inf)

concave up (-4, 4)

x = 4 and x = -4 are IP.

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