I have a function of $f(x)=x^5+x$ on the interval $[-2,2]$.
By using the first order derivative method of convexity we have $f(x) ≥f(x_0)+∇f(x_0)^T(x-x_0)$ and on the right hand side I get $-4x_0^5+5x_0^4+x$ from which we can't tell whether it's bigger or smaller than $x^5+x$.
I tried the second order derivative method as well, which states that $f(x)$ is convex if and only if $f''(x)≥0$ but the second order derivative of our function yields $20x^3$ which can be both negative and positive on our interval.
So does it mean that the function is neither convex nor concave?