Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 1: EXERCISE-1

Let  $\left\{\begin{array}{ccc}\frac{4-x}{2-\sqrt{x}}& \mathrm{for}& 0<x<4\\ 4& \mathrm{for}& x-4\\ 16-3x& \mathrm{for}& 4<x<6\end{array}\right.$

Which of the following properties does $f$ have on the interval $\left(0,6\right) ?$

(I) $\ln \left(f\left(x\right)\right)$ exists

(II) $f$ is continuous

(III) $f$ is monotonic

The largest set of real values of $x$ for which $\ln \left(1+x\right)\le x$ is

Number of solution of the equation $3\tan x+x^3=2$ in $\left(0,\frac{\pi }{4}\right)$ is

Two sides of a triangle are to have lengths '$a$' $\mathrm{cm}$ and '$b$' $\mathrm{cm}$. If the triangle is to have the maximum area, then the length of the median from the vertex containing the sides '$a$' and '$b$' is -

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

A solid rectangular brick is to be made from $1 cu$ feet of clay. The brick must be $3$ times as long as it is wide. The width of brick for which it will have minimum surface area is $a$ feet. Then $a^3$ is -

Which of the following statements is true for the general cubic function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$, $\left(a\ne 0\right)$.

$\mathrm{I}$. If the derivative $f^{\text{'}}\left(x\right)$ has two distinct real roots, then cubic has one local maxima and one local minima.

$\mathrm{II}$. If the derivative $f^{\text{'}}\left(x\right)$ has exactly one real root, then the cubic has exactly one relative extrema.

$\mathrm{III}$. If the derivative $f^{\text{'}}\left(x\right)$ has no real roots, then the cubic has no relative extrema

The gradient of the tangent line at the point $\left(a\cos \alpha ,a\sin \alpha \right)$ to the circle $x^2+y^2=a^2$, is