If `f` is an even function such that `(lim)_(hvec0)(f(h)-f(0))/h` has some finite non-zero value, then prove that `f(x)` is not differentiable at `x=0.`

(lim)_(hvec0)((e+h)^(1n(e+h))-e)/hi s____

Which of the function is non-differential at x=0? f(x)=|"x"|

Let f(x) be a function defined on (-a ,a) with a > 0. Assume that f(x) is continuous at x=0a n d(lim)_(xvec0)(f(x)-f(k x))/x=alpha,w h e r ek in (0,1) then f^(prime)(0^+)=0 b. f^(prime)(0^-)=alpha/(1-k) c. f(x) is differentiable at x=0 d. f(x) is non-differentiable at x=0

Which of the function is non-differential at x=0? f(x)="cos"|x|

If |f(x)|lex^(2), then prove that lim_(xto0) (f(x))/(x)=0.

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

If f:RR-> RR is a differentiable function such that f(x) > 2f(x) for all x in RR and f(0)=1, then

If f(x) is a twice differentiable function such that f(a)=0, f(b)=2, f(c)=-1,f(d)=2, f(e)=0 where a < b < c < d e, then the minimum number of zeroes of g(x) = f'(x)^2+f''(x)f(x) in the interval [a, e] is

If f(x)=2x^(2)+3x-5 , then prove that f'(0)+3f'(-1)=0

Let f(x),xgeq0, be a non-negative continuous function, and let F(x)=int_0^xf(t)dt ,xgeq0, if for some c >0,f(x)lt=cF(x) for all xgeq0, then show that f(x)=0 for all xgeq0.