Question

Is `f(x)=3x^5-x^3+8x^2-x` concave or convex at `x=0`?

Answer 1

`f(x)` is concave at `x=0.`

Explanation:

`f(x)` is concave at `x=a` if `f''(a)>0`.

`f(x)` is convex at `x=a` if `f''(a)>0`.

Here, `f(x)=3x^5-x^3+8x^2-x, a=0.` Knowing that, let's take the second derivative.

`f'(x)=15x^4-3x^2+16x-1`

`f''(x)=20x^3-6x+16`

Evaluate at `0:`

`f''(0)=20(0^3)-6(0)+16=16>0`

`f(x)` is concave at `x=0.`