Embibe Experts Solutions for Exercise 7: Assignment

Consider the functions $f\left(x\right)=\left\{\begin{array}{l}\left|x\right|, x\le -1\\ x^{\frac{1}{5} }-1<x\le 1\\ {\left(2-x\right)}^3, x>1\end{array}\right.$ and $g\left(x\right)=\frac{1}{{\left(x+2\right)}^3}-2x-\cos x$. Let $p$ be the number of critical points on the graph of $f\left(x\right)$ and $q$ be the number of solutions of $g\left(x\right)=0$, then which of the following option(s) is/are correct?

Let $A$ be the maximum area of the incircle of the right-angled triangle with hypotenuse of length $\left(\sqrt{2}+1\right)$ units and $B$ be the minimum area of the circumcircle of the right-angled triangle with area $4$ square units. Then, which of the following option(s) is/are correct?

The length of the portion of a tangent line to the curve $x^{2/3}+y^{2/3}=4$ at the point $P$, cut off by the coordinate axes, is

Which of the following statement(s) is/are correct?

Which of the following statement(s) is/are correct for the function $f\left(x\right)=\frac{x^2-2}{3x^2-9}$?

Any tangent at a point $P\left(x, y\right)$ to the ellipse $\frac{x^2}{8}+\frac{y^2}{18}=1$ meets the positive co-ordinate axes in the points $A$ and $B$ such that the area of the $∆OAB$ is least, then the point $P$ is of the form $\left(m, n\right)$ where $m+n+10$ is

If the tangent at $P\left(1,1\right)$ on the curve $y^2=x{\left(2-x\right)}^2$ meets the curve again at $Q$, then the point $Q$ is of the form $\left(\frac{3a}{b},\frac{a}{2b}\right)$, where minimum integral value of $a^2+b^2$ is

The curve $y=a{x}^3+b{x}^2+cx$ is inclined at $45°$ to $x$-axis at $\left(0,0\right)$ but it touches $x$-axis at $\left(1,0\right)$, then $a+b+c+10$ is