Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 1: JEE Advanced Paper 1 - 2021

Let the function $f:\left(0,\pi \right)\to \mathrm{ℝ}$ be defined by $f\left(\theta \right)={\left(\sin \theta +\cos \theta \right)}^2+{\left(\sin \theta -\cos \theta \right)}^4$. Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \left\{{\lambda }_1\pi ,\dots ,{\lambda }_r\pi \right\},$ where $0<$${\lambda }_1<\cdots <{\lambda }_r<1.$ Then the value of  ${\lambda }_1+\cdots +{\lambda }_r$ is_______

Let $f\left(x\right)=\frac{\sin \pi x}{x^2}, x>0.$
Let $x_1<x_2<x_3<\dots <x_n<\dots$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<\dots <y_n<\dots$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?

Let $f:R\to R$ be given by $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-5\right).$ Define $F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt, x>0.$ Then which of the following options is/are correct?

If $f:R\to R$ is a differentiable function such that $f^{\prime }\left(x\right)>2f\left(x\right)$ for all $x\in R$ and $f\left(0\right)=1$ then

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$

Then which of the following statements is(are) TRUE?

Consider all rectangles lying in the region $\left\{\left(x, y\right)\in R\times R:0\le x\le \frac{\pi }{2}and0\le y\le 2\sin \left(2x\right)\right\}$ and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

Let $T$ denote a curve $y=y\left(x\right)$ which is in the first quadrant and let the point $\left(1,\mathrm{ }0\right)$ lie on it. Let the tangent to $T$ at a point $P$ intersect the $y$-axis at $Y_P.$ If $P{Y}_P$ has length $1$ for each point $P$ on $T,$ then which of the following options is/are correct?

Let $f\left(x\right)=x+{\log }_{e}x-x{\log }_{\mathrm{e}}x,x\in \left(0, \infty \right).$

$•$   Column 1 contains information about zeros of $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right).$

$•$   Column 2 contains information about the limiting behaviour of $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ at infinity.

$•$   Column 3 contains information about increasing-decreasing nature of $f\left(x\right)$ and $f^{\prime }\left(x\right).$