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

Find the equation of tangent and normal to the curve $x^2+3xy+y^2=5$ at point $(1,1)$ on it.

Find the equation of tangent and normal to the curve $x=\frac{2a{t}^2}{1+t^2},y=\frac{2a{t}^3}{1+t^2}$ at the point for which $t=\frac{1}{2}$.

If the tangent at $(1,1)$ on $y^2=x(2-x{)}^2$ meets the curve again at $P$, then find coordinates of $P$

The normal to the curve $5x^5-10x^3+x+2y+6=0$ at the point $P(0,-3)$ is tangent to the curve at some other point(s). Find those point(s)?

Find common normals of the curves $y=\frac{1}{x^2}$ and $x^2+y^2-y=0$

The curve $y=a{x}^3+b{x}^2+3x+5$ touches $y=(x+2{)}^2$ at $(-2,0)$ then $\left|\frac{a}{2}+b\right|$ is

Let $f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x$ be an increasing function in the set of real numbers $R$. Then $a \& b$ satisfy the condition:

Which of the following statement is/are true ?

(1) $f\left(x\right)=\sin x$ is increasing in interval $\left[\frac{-\pi }{2},\frac{\pi }{2}\right.⌋$
(2)$f\left(x\right)=\sin x$ is increasing at all point of the interval $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$
(3) $f\left(x\right)=\sin x$ is increasing in interval $\left(\frac{-\pi }{2},\frac{\pi }{2}\right)\cup \left(\frac{3\pi }{2},\frac{5\pi }{2}\right)$
(4) $f\left(x\right)=\sin x$ is increasing at all point of the interval $\left(\frac{-\pi }{2},\frac{\pi }{2}\right)\cup \left(\frac{3\pi }{2},\frac{5\pi }{2}\right)$
(5) $f\left(x\right)=\sin x$ is increasing in intervals $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]\cup \left[\frac{3\pi }{2},\frac{5\pi }{2}\right]$