If $f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3)\,dt$ for all $x \in (0,\infty)$, then

If f(x)=∫₀ˣ e^(t²)(t-2)(t-3) dt for all x ∈ (0,∞), then $f(x) = \int_0^x e^{t^2}(t-2)(t-3)\,dt, \qquad x \in (0,\infty)$ $\text{Which of (1)-(4) are TRUE?}$ $1.\ f'(x) \text{ by FTC}$ $2.\ \text{sign of } f' \Rightarrow \text{extrema, monotonicity}$ $3.\ f''(x) = 0 \text{ for some } c$ $f'(x) = e^{x^2}(x-2)(x-3) \quad (\text{FTC})$ $e^{x^2} > 0 \Rightarrow \operatorname{sign} f' = \operatorname{sign}\big[(x-2)(x-3)\big]$ $f' > 0 \text{ on } (0,2)\cup(3,\infty), \qquad f' < 0 \text{ on } (2,3)$ $(1):\ f' \text{ changes } + \to - \text{ at } x=2 \Rightarrow \text{local maximum} \Rightarrow \text{TRUE}$ $(2):\ f' < 0 \text{ on } (2,3) \Rightarrow f \text{ decreasing on } (2,3) \Rightarrow \text{TRUE}$ $(4):\ f' \text{ changes } - \to + \text{ at } x=3 \Rightarrow \text{local minimum} \Rightarrow \text{TRUE}$ $f''(x) = e^{x^2}\big[2x(x-2)(x-3) + (2x-5)\big] = e^{x^2}\big[2x^3-10x^2+14x-5\big]$ $p(0) = -5 < 0, \quad p(1) = 1 > 0 \Rightarrow \exists\, c \in (0,1): f''(c) = 0 \Rightarrow \text{TRUE}$ $\boxed{\text{All four } (1), (2), (3), (4) \text{ are TRUE}}$