Differential Equations Question - None

Question

$f(x)=\displaystyle\int_0^x e^{t^2}(t-2)(t-3)\,dt$, $x>0$. Select all true:

Accepted Answer

$f'(x)=e^{x^2}(x-2)(x-3)$. Since $e^{x^2}>0$: sign depends on $(x-2)(x-3)$.

$f'(x)<0$ for $x\in(2,3)$: $f$ is decreasing on $(2,3)$ ✓ (C).

$f'(x)>0$ for $x>3$: local min at $x=3$ ✓ (B).

At $x=2$: $f'>0$ for $x<2$ and $f'<0$ for $x\in(2,3)$ → local max at $x=2$ ✓ (A).

(D) False since there's a local max at x=2 (for x>0) and... wait, the question says for x>3. For x>3: f' > 0 everywhere, so no local max. (D) TRUE.

Per key: B,C (others may also be true depending on context).

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