MathNet 1998 — Part 01 Question 13

Let $P$ be a polynomial of degree $6$ and let $a, b$ be real numbers such that $0 \lt a \lt b$. Suppose that $P(a) = P(-a)$, $P(b) = P(-b)$ and $P'(0) = 0$. Prove that $P(x) = P(-x)$ for all real $x$.