미적분의 기본정리

By the Fundamental Theorem of Calculus Part 1,

$$f'(x) = 100(x^2-3x+2)e^{-x^2} = 100(x-1)(x-2)e^{-x^2}$$

As $f(x)$ is increasing whenever $f'(x)>0$ and $100 e^{-x^2}$ is always strictly bigger than $0$, we have $f(x)$ increasing if and only if $(x-1)(x-2)>0$, which is the case if and only if $(x-1)$ and $(x-2)$ are of the same sign. Both are positive when $x>2$ and both are negative when $x<1$. So $f(x)$ is increasing when $-\infty<x<1$ and when $2<x<\infty$.

Remark: even without the Fundamental Theorem of Calculus, since $f(x)$ is the area under a curve from 1 to $x$, $f(x)$ is increasing when the curve is above the $x$-axis (because we're adding positive area), and it's decreasing when the curve is below the $x$-axis (because we're adding negative area).