Question

g(x)=x3−12x2+45x+8 d) On what interval(s) is g concave up or concave down? Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is concave up on and concave down on B. The function is concave up on and is never concave down. C. The function is concave down on and is never concave up. e) Choose the correct graph below. A. B. = D.

Answer 1

Concave down on
`(-\infty,4)`

Concave up on
`(4,\infty)`

Explanation:

Determine possible inflection points

`g(x)=x^3-12x^2+45x+8\\g'(x)=3x^2-24x+45\\g''(x)=6x-24\\\\0=6x-24\\24=6x\\x=4`

Use test points

`g''(3)=6(3)-24=-6 < 0`, so
`g(x)` is concave down over
`(-\infty,4)`

`g''(5)=6(5)-24=6 > 0`, so
`g(x)` is concave up over
`(4,\infty)`