Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 4: Exercise-4

Prove the following inequalities $\sin x-\sin 2x\le 2x$ for all $x\in \left[0,\frac{\pi }{3}\right]$

Prove the following inequalities $\frac{x^2}{2}+2x+3\ge \left(3-x\right)e^x$ for all $x\ge 0$

Prove the following inequalities $0<x\sin x-\frac{{\sin }^{2}x}{2}<\frac{1}{2}\left(\pi -1\right)$ for $0<x<\frac{\pi }{2}$

Show that the volume of the greatest cylinder which can be inscribed in a cone of height $h$ and semi-vertical angle $\alpha$ is $\frac{4}{27}\pi h^3{\tan }^2\alpha$..

Let $f^{\text{'}}\left(\sin x\right)<0$ and $f^{"}\left(\sin x\right)>0,\forall x\in \left(0,\frac{\pi }{2}\right)$ and $g\left(x\right)=f\left(\sin x\right)+f\left(\cos x\right),$ then find the intervals of monotonicity of $g\left(x\right).$

Find the set of values of the parameter $\text{'}a\text{'}$ for which the function; $f\left(x\right)=8ax-a\sin 6x-7x-\sin 5x$ increases & has no critical points for all $x\in R,$ is

Let $h$ be a twice differentiable positive function on an open interval $R$. Let $g\left(x\right)=\ln \left(h\left(x\right)\right)\forall x\in R$
Suppose ${\left(h^{\text{'}}\left(x\right)\right)}^2>h^{"}\left(x\right)h\left(x\right)$ for each $x\in R$. Then prove that $g$ is concave downward on $R$

If two curves $y=2\sin \frac{5\pi }{6}x$ and $y=\alpha x^2-3\alpha x+2\alpha +1$ touch each other at some point then the value of $\frac{\sqrt{3}\alpha }{5\pi }$ is