Question

Determine whether f is differentiable at the given point. If f'(a) exists, give its value. HINT [See Example 3.] (If an answer does not exist, enter DNE.) f(x) = x1/5 + 6 (a) a = 32 (b) a = 0

Answer 1

Your question is solved step by step & is given below.

Ls Solution: f(x) = x + 6 (a) At a=32 Left Hand Differentiability lim ho h f(32) = löm f122) = f(3=8) [t'an-fion for forte) = Lim (534x8= (32-0)36 (Vinga) - dom wat [ 2-(4-81 = lim  [=(1-4 6--)) ht N *32 = 2*2 = bo RHD f (92) = dem f(32+8)= f(32) - lim (32th 15 + 6-{(32) 5 +67 htoh 121175 = (Same as above) f-(32) = f*(32) = So, 80

Mence, at a=32, f'(a) exists and (4) When a=0 Left Hand Derivative (LHD) t'10) = lim f(o)-flock huo hd = lim &- lchfs ! (:f101= 0 +6=6) htoh lim -(-4) 15 & 1/5 ht0 h lim - (-1) 15 N (ONE) to 4415 Right Hand Derivative (RHD) f(0) = lim floth)-f(o) ho h. - lim 615+6-6 to h = lim ola - - (ONE) ho h Thes, at a=0, fla) does not exist (DNE). Alternatively, f(x)= x^5 +6 z f'(X) = x 45- 46 So, If (22) = 1 and (f(0) = + gie, ONE &
Two binomial formulae are used in above solution, which are :

nin 24 It nyt X 21 © (1+x)" = 1+nx + n(n-1) x2 + n(n-1)(-2)x+ --- (1-x) = 1- nx + n(n+1)x?_n (n=1/M-) +--- - 2 han h - - NX+ 2

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