5) Check whether f(t)= { Х Bing if x=0 el if x=0 differentiable at x =...

First we for ME [0, T/). prone that ne tanu Define! f(n):= na tana Casual f'(W) = 1- Costa 1 = Cosu since n co [-(Cosule i = f(n) is monetonie decreasing, and f(0) = 0, we get that for ne 2 E [0, T42), u-tanu = fu) <fo) = 0 =) ne tanu for ne [0, 7/2). Therefore, na tann for ne for he [0, T/ This implies that Cosh e Sinu for he [0, 7/2) Also we know that Sintax for Also, similarly we can prove Sinu su for ne [O, TY2) ie, Sinu al for at [0, 1/2) Therefore by combining , we get Cosna for ne {0, 72) Similashy we Cosa s fer n E (-12,0] prove that Sinu L 1 can prane Sinu ci

take d=26 LT/2. Hence, we h - Sinu - Casual < 2 get Cosh < Sinu al for ne (7 7 ) or Ink% Siam leo I Sinu -l | <a (sinh 1 22.11 for Ink TL - ".. Sinual we We 5 £ Inl 70 prove that f(n) is differentiable at hao and f(0) = 0.

To prove this need to show that for any EXO, 7 a do sit. whenever In-oko, 2 f(n) - f(0) - oke n-o Let us choose Exo sit. E<T/4 . Then inte con Therefore from (1), we 20 1 S 11 = { 121 Lt whenever In-oko. Sinu 0 n Therefore, N-o differentiable and f'(o)=0.

Hence, f(n) is