Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 1: VITEEE 2018 Shift 1

The radius of a right circular cylinder increases at the rate of $0.1 \mathrm{cm}/\min$, and the height decreases at the rate of $0.2 \mathrm{cm}/\min$. The rate of change of the volume of the cylinder in ${\mathrm{cm}}^3/\min$, when the radius is $2 \mathrm{cm}$ and the height is $3 \mathrm{cm}$ is

The volume $V$ and depth $x$ of water in a vessel are connected by the relation $V=5x-\frac{x^2}{6}$ and the volume of water is increasing at the rate of $5{\mathrm{cm}}^3/\sec .$ When $x=2\mathrm{cm}$, the rate at which the depth of water is increasing is

The angle between a pair of tangents drawn from a point $T$ to the circle $x^2+y^2+4x-6y+9{\sin }^2\alpha +13{\cos }^2\alpha =0$ is $2\alpha .$ The equation of the locus of the point $T$ is

If $f\left(x\right)=x^3+b{x}^2+cx+d$ and $0<b^2<c$, then in $(-\infty ,\infty )$

The function $f\left(x\right)=\sin x-kx-c$, where $k$ and $c$ are constants, decreases always when

A wire $34 \mathrm{cm}$ long is to be bent in the form of a quadrilateral of which each angle is ${90}^{°}$. What is the maximum area which can be enclosed inside the quadrilateral?

The difference between the greatest and the least value of $f\left(x\right)=2\sin x+\sin 2x, x\in \left[0,\frac{3\pi }{2}\right]$ is

The value of $x$ in the interval $[4,9]$ at which the function $f\left(x\right)=\sqrt{x}$ satisfies the mean value theorem is