Let $f: R \to R$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define $F(x)=\int_0^x f(t)\,dt, x>0$. Then which of the following options is/are correct?

Let f: R → R be given by f(x)=(x-1)(x-2)(x-5). Define F(x)=∫₀ˣ f(t) dt, x>0. Then which of the following options is/are correct?

$f(x)=(x-1)(x-2)(x-5)$

$F(x)=\int_0^x f(t)\,dt,\quad x>0$

$\text{To find: correct statement(s) about } F$

$F'(x)=f(x)$ $f(x)=0 \Rightarrow x=1,\ 2,\ 5 \text{ are critical points of } F$ $f<0\ (0,1);\quad f>0\ (1,2);\quad f<0\ (2,5);\quad f>0\ (5,\infty)$ $x=1:\ \min;\quad x=2:\ \max;\quad x=5:\ \min$ $F(x)=\dfrac{x^4}{4}-\dfrac{8x^3}{3}+\dfrac{17x^2}{2}-10x$ $F(2)=4-\dfrac{64}{3}+34-20=-\dfrac{10}{3}<0$ $F(0)=0,\ F(2)=-\tfrac{10}{3} \Rightarrow F(x)<0\ \forall x\in(0,5)$ $\text{(1) true, (2) true, (3) false, (4) true}$ $\boxed{\text{options (1), (2), (4)}}$