2019
Problem - 4532
Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:
- it is an even function
- $f(2)=f(-2)=0$
- $f(x) > 0$ when $-2 < x < 2$, and
- the maximum of $f(x)$ is achieved at $x=\pm 1$.
Because $f(x)$ is even and $f(x) > 0$ in $(-2, 2)$, therefore $f(x)$ has a non-negative local minimal at $x=0$. Therefore, $f'(x)$ has at least three zeros: $0$ and $\pm 1$. Let
$$f'(x)=x(x+1)(x-1)=x-x^3\implies f(x) = \frac{x^2}{2}-\frac{x^4}{4} + C$$
Letting $f(\pm 2)=0$ leads to $C=2$. Then it is easy to verify that the following polynomial satisfies all the constraints:
$$f(x)=\boxed{\frac{x^2}{2}-\frac{x^4}{4}+2}$$