Question

Check the differentiability of the function f(x)=x^2*{|cos(pi/x)|} at x=2

Answer 1

First\[|x|=\sqrt{x^2}\]
\[f(x)=x^2{|\cos(\frac{\pi}{x})|}=(x \sqrt{\cos(\frac{\pi}{x})} )^2 \]

Answer 2

well. I was trying use the formula F'(x)= lim h--->{2f(x+2) -f(h)}/h, but I messed up!
would you help me doing that?

Answer 3

\[f'(x)=\lim(h \rightarrow 0)(\frac{f(x+h)-f(x)}{h})\]

Answer 4

yeah

Answer 5

This function is not differentiable at x=2. The right hand derivative is not equal to the left hand derivative.

Answer 6

If x < 2 and close to 2
\[
\cos(\pi/x) <0
\]
If x > 2 and close to 2
\[
\cos(\pi/x) >0
\]

Answer 7

The derivative a left of 2 is
\[
u(x)=-\pi \sin \left(\frac{\pi
}{x}\right)-2 x \cos
\left(\frac{\pi }{x}\right)\\
v(2)=-\pi
\]
The derivative a right of 2 is
\[
v(x)=\pi \sin \left(\frac{\pi
}{x}\right)+2 x \cos
\left(\frac{\pi }{x}\right)\\
u(2)=\pi
\]