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

The slope of the tangent to the curve ${\left(y-x^5\right)}^2=x{\left(1+x^2\right)}^2$ at the point $(1,3)$ is

The number of tangents to the curve $y=\cos (x+y),-2\pi \le x\le 2\pi$, that are parallel to the line $x+2 y=0$ is

If the curves $x=y^4$ and $xy=k$ cut at right angles, then ${\left(4k\right)}^6$ is equal to ___ .

Which one of the following is correct in respect of the function $f\left(x\right)=x\left(x-1\right)\left(x+1\right)?$

Let $n$ be the number of critical points of $f\left(x\right)=\frac{{\left(\ln x\right)}^2}{x}$, then $4\left(n^2+\frac{1}{n^2}\right)=$

If the minimum value of $f\left(x\right)=\frac{64}{\cos x}+\frac{27}{\sin x}$, where $x\in \left(0, \frac{\mathrm{\pi }}{2}\right)$, is equal to $\lambda$, then the value of $\lambda$ is 

If $f\left(x\right)=\left\{\begin{array}{l}{e}^x\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\infty ,<x\le 1\\ 2-e^{x-1} ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1<x\le 2\\ x-e\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2<x<\infty \end{array},\right.$ then the sum of value of $f\left(x\right)$ at the point of local maximum and local minimum is