Suppose the function g(x) has a domain of all real numbers. The second derivative of g(x) is shown below.

Question

g ''(x) = (x−2)⁵(x + 3)(x + 4)²

a) Give the intervals where g(x)

 is concave up. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

(b) Give the intervals where g(x)

 is concave down. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

(c) Find the x-coordinates of the inflection points for g(x)

. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x = 

Answer 1

Concave up where g''(x) > 0

Draw function on Desmos and read off the intervals.

Alternatively : the zeroes are 2, -3, -4

so evaluate g'' in the intervals (- infinity, -4) (-4, -3) (-3, 2) and (2, infinity) and choose the intervals where g'' > 0

Concave down

g'' (x) < 0 - use ideas from above

Points of inflection : g'' =0 , and g'' changes sign via the point

so x = 2, -3, -4 are possible

check if g''(x) changes sign through those points - multiplicities of the zeroes tells you or you can look at your analysis from the first part