Question

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

Answer 1

convex at x = 2

Explanation:

To determine if a function f(x) is concave/convex at x = a we require to find the value of f''(a).

• If f''(a) > 0 then f(x) is convex at x = a

• If f''(a) < 0 then f(x) is concave at x = a

Now `f(x)=x^3-2x+7`

differentiate using the `color(blue)"power rule"`

`rArrf'(x)=3x^2-2`

and `f''(x)=6x`

`rArrf''(2)=(6xx2)=12`

Since f''(2) > 0 then f(x) is convex at x = 2.
graph{x^3-2x+7 [-22.5, 22.5, -11.25, 11.25]}