Suppose is a twice differentiable function with and . Let be defined by
if , if
Prove that is differentiable at and find .
Can we use L' Hopital's Rule to solve this question?
Use L' Hopital's Rule to find the limit of f(x) at x=0 first.
f(x) = g(x)/x, since g is a differentiable function,
therefore f(x) must be differentiable somewhere on R.
assume f'(x) exists at point x=k, then
f'(k) = [k g'(k) - g(k) ]/k^2. (by quotient rule)
take above expression become a limit which k approach 0 and apply L' Hopital's Rule to show that f is differentiable at x=0, and hence find f'(0), the answer should be 7.