Question

Is `f(x) =2x^3-(x+2)(x+7)` concave or convex at `x=-8`?

Answer 1

Concave.

Explanation:

Convexity and concavity can be found through the second derivative:

If `f''(-8)>0`, then the function is convex when `x=-8`.
If `f''(-8)<0`, then the function is concave when `x=-8`.

To find the second derivative of the function, first simplify `f(x)`.

`f(x)=2x^3-(x^2+9x+14)`
`f(x)=2x^3-x^2-9x-14`

Now, find the second derivative.

`f'(x)=6x^2-2x-9`
`f''(x)=12x-2`

Now calculate `f''(-8)`.

`f''(-8)=12(-8)-2=-98`

Since `-98<0`, the function is concave when `x=-8`.

We can reference a graph—a concave shape should resemble the `nn` shape at `x=-8`.

graph{2x^3-(x^2+9x+14) [-10, 5, -1500, 500]}