Question 15
Find all fourth degree polynomials $p(x)$ such that the following four conditions are satisfied:
(i) $p(x) = p(-x)$ for all $x$.
(ii) $p(x) \geq 0$ for all $x$.
(iii) $p(0) = 1$.
(iv) $p(x)$ has exactly two local minimum points $x_1$ and $x_2$ such that $|x_1 - x_2| = 2$.
(i) $p(x) = p(-x)$ for all $x$.
(ii) $p(x) \geq 0$ for all $x$.
(iii) $p(0) = 1$.
(iv) $p(x)$ has exactly two local minimum points $x_1$ and $x_2$ such that $|x_1 - x_2| = 2$.