Question

Show that the function $f\left(x\right)=|x-3|,xϵR$ is continuous but not differentiable at $x=3$ Or $x=asintandy=a(cost+log\left(tan\frac{t}{2}\right))$

find $\frac{d^{2}y}{d{x}^2}$

Answer 1

$f\left(x\right)=\{\begin{array}{c}3-x;x<3\\ 0;x=3\\ x-3;x>3\end{array}$

${lim}_{x\to 3^{-}}f\left(x\right)={lim}_{x\to 3^{-}}(3-x)=0$

${lim}_{x\to 3^{+}}f\left(x\right)={lim}_{x\to 3^{+}}(x-3)=0$

LHL$=$RHL $=f\left(0\right)$

Continuous at $x=3$

$f^{\prime }\left(x\right)=\{\begin{array}{c}-1;x<3\\ 0;x=3\\ 1;x>3\end{array}$

${lim}_{x\to 3^{-}}{f}^{\prime }\left(x\right)=-1$

${lim}_{x\to 3^{+}}{f}^{\prime }\left(x\right)=1$

LHL$\ne$RHL

Not differentiable at $x=3$

(OR)

$x=a\sin t,y=a(\cos t+\log \left(\tan t/2\right))$

$y^{\prime }=\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$

$\frac{dy}{dt}=a(-\sin t+\frac{{\sec }^{2}t/2}{\tan t/2}\times \frac{1}{2})$$=a(-\sin t+\frac{1}{\sin t})$

$=a\left(\frac{1-{\sin }^{2}t}{\sin t}\right)=a\left(\frac{{\cos }^{2}t}{\sin t}\right)$

$\frac{dx}{dt}=a\cos t$

$y^{\prime }=\frac{a{\cos }^{2}t/\sin t}{a\cos t/1}=\cot t$

$\frac{d^{2}y}{d{x}^2}=\frac{d{y}^{{}^{\prime }}/dt}{dx/dt}$

$\frac{d{y}^{{}^{\prime }}}{dt}=-{\csc }^{2}t,\frac{dx}{dt}=a\cos t$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{-{\csc }^{2}t}{a\cos t}$