OpenStudy (anonymous):

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

6 years ago
OpenStudy (anonymous):

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

6 years ago
OpenStudy (anonymous):

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?

6 years ago
OpenStudy (anonymous):

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

6 years ago
OpenStudy (anonymous):

yeah

6 years ago
OpenStudy (anonymous):

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

6 years ago
OpenStudy (anonymous):

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

6 years ago
OpenStudy (anonymous):

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 \[ v(x)=\pi \sin \left(\frac{\pi }{x}\right)+2 x \cos \left(\frac{\pi }{x}\right)\\ u(2)=\pi \]

6 years ago