Proof of $n^{th}$ derivative Test Proof needn't be a rigourous , but should give an insight of how $n^{th}$ derivative test (higher order derivative test) works as i know how to use it in application but i don't much understand it ,especially the inflexion point thing.

Let $f$ be $C^k$ (i.e., differentiable $k$ times with continuous $k$th derivative; we can relax this condition a bit, but it's not really useful here) on a neighborhood of a point $x$ , which we'll assume without loss of generality is $0$. Then by Taylor's theorem, we have
$$f(x) = f(0) + f'(0)x + \frac{1}{2}f''(0)x + \cdots + \frac{1}{k!} f^{(k)}(0) x^k + o(x^k)$$
near $x = 0$. If the first $(k-1)$st derivatives vanish then we're left with
$$f(x) = \frac{1}{k!}f^{(k)}(0) x^k + o(x^k).$$
Near $0$, the term $\frac{1}{k!} f^{(k)}(0) x^k$ dominates the sum above. If $k$ is even, then $x^k$ has a minimum at $x = 0$, which turns into a minimum or maximum of $f$ depending on the sign of $f^{(k)}(0)$. If $k$ is odd, then $f$ behaves like $x^k$ near $0$, which has neither a minimum nor a maximum at $x = 0$.
