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Question:

Let f:[a,b]→[1,∞) be a continuous function and let g:R→R be defined as g(x) = {0 if x<a, ∫xaf(t)dt if a≤x≤b, ∫baf(t)dt if x>b. Then

g(x)is differentiable onR

g(x)is continuous but not differentiable atb

g(x)is continuous and differentiable at eitheraorbbut not both

g(x)is continuous but not differentiable ata

Solution:

Since f(x)≥1 ∀x∈[a,b] for g(x) LHD at x=a is zero and RHD at (x=a)=limx→a+∫xaf(t)dt/(x−a)=limx−a+f(x)≥1
Hence, g(x) is not differentiable at x=a
Similarly LHD at x=b is greater than 1
g(x) is not differentiable at x=b