$\max_{0 \leq x \leq \pi}\mspace{2mu}\left( 16sin\left( \frac{x}{2} \right)\cos^{3}\left( \frac{x}{2

Question

$\max_{0 \leq x \leq \pi}\mspace{2mu}\left( 16sin\left( \frac{x}{2} \right)\cos^{3}\left( \frac{x}{2} \right) \right)$ is equal to :

Accepted Answer

Sol. [Key Concept: Write $E = 4sin\ x(1 + cos\ x)$; set $E' = 0$, then evaluate $E$ at the critical point $x = \pi/3$.]$${E = 16sin\frac{X}{2}\cos^{3}\frac{X}{2} }{E = 4sinx\lbrack 1 + cosx\rbrack }{\frac{dE}{dx} = 4\lbrack cosx + cos2x\rbrack }{= 8cos\frac{3x}{2}cos\frac{x}{2} = 0 }$$$\Rightarrow cos\frac{3x}{2} = 0\ $ or $\ cos\frac{x}{2} = 0$$\Rightarrow x = \left\{ \frac{\pi}{3},\pi \right\}$ are critical points of the function$${E(0) = 0 }{E(\pi) = 0 }$$E $\left( \frac{\pi}{3} \right) = 3\sqrt{3}$∴ maximum value of $E = 3\sqrt{3}$

Question

Options

Solution

More Application of Derivative Questions