Maximum and minimum for piecewise function

Question

I am tasked with finding the exact value for the maximum and minimum of the function

$f(x) = \begin{cases} 0 & x \leq 1001 \\ \exp(- \frac{1}{(-1001+x)^2(-1003+x)^2}) & 1003 > x > 1001 \\ 0 & x \geq 1003\end{cases}$

It is fairly easy to see that a minimum is $x = 0$, but as far as I can tell there is no exact value for a maximum. There is however a supremum, but that is not what the question asks.

Am I missing something, or is this question misleading?

Answer 1

Actually, there is a maximum, not just a supremum. The maxmimum is obtained in $x=1002$.

This is because $x=1002$ maximizes $(-1001+x)^2(-1003+x)^2$ on the interval $(1001,1003)$, and therefore maximizes $\exp(-\frac1{(-1001+x)^2(-1003+x)^2})$.

Answer 2

The maximum occurs at the minimum of the fraction (minus sign omitted), which occurs at the maximum of the denominator, which occurs at the maximum of the product $|(-1001+x)(-1003+x)|$, which occurs at $1002$.