$\max _{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$

If the distance $s$ covered in time $t$ by a particle moving on a straight line is given by $s = t^5 - 40t^3 + 30t^2 + 80t - 250$,then its minimum acceleration is

Find the maximum area of an isosceles triangle inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with its vertex at one end of the major axis.

The perimeter of a sector is $p$. What is its radius when the area of the sector is maximum?

The maximum value of the function $f(x) = 3x^3 - 18x^2 + 27x - 40$ on the set $S = \{x \in R : x^2 + 30 \leq 11x\}$ is

Let $f: R \rightarrow R$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define $F(x)=\int_0^x f(t) dt$ for $x>0$. Which of the following options is/are correct?
$(1)$ $F$ has a local minimum at $x=1$
$(2)$ $F$ has a local maximum at $x=2$
$(3)$ $F(x) \neq 0$ for all $x \in (0,5)$
$(4)$ $F$ has two local maxima and one local minimum in $(0, \infty)$