The answer is 225.

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

The function must be odd, so that $f(0) = 0$ .

If $f(|x|) = |f(x)|$ then $f(x) > 0$ for all positive $x$ .

We are told that $f(4) = 0$ but because $f(x)$ does not become negative in the neighborhood of $x = 4$ , this must be a root of even multiplicity. Because of symmetry, $-4$ is also a root of even multiplicity.

Now we know at least five roots: $-4, -4, 0, 4, 4$ . There cannot be any other. Thus $f(x) = a(x+4)^2\:x\:(x-4)^2.$ Since the polynomial is monic, $a = 1$ . Finally $f(1) = (1+4)^2\cdot 1\cdot (1-4)^2 = 25\cdot 1\cdot 9 = \boxed{225}.$