Question

Is `f(x)=-2x^5+7x^2+8x-13` concave or convex at `x=-4`?

Answer 1

Since `f''(-4) > 0`, `f(x)` is convex at `x = -4`.

Explanation:

Let's compute the second derivative:

`f(x) color(white)(ii) = -2x^5 + 7x^2 + 8x - 13`
`f'(x) color(white)(i) = -10 x^4 + 14 x + 8`
`f''(x) = -40 x^3 + 14`

Now, let's evaluate the second derivative at `x = -4` and check if `f''(-4)` is negative or positive:

`f''(-4) = -40 * (-4)^3 + 14`

Even without computing this value, you can see that `(-4)^3` is negative and `-40` is also negative. A negative value multiplied with a positive value is positive. `14` is positive as well.

So we can conclude that

`f''(-4) > 0`

This means that `f(x)` is convex at `x = -4`.