Question

Let f:[a, b] → [1, ∞) be a continuous function and g:R → R be defined as

g(x) = {(0 : x < a), (∫f(t)dt for t ∈ [a, x] : x ≤ x ≤ b), (∫f(t)dt for t ∈ [a, b] : x > b)

(a) g(x) is continuous but not differentiable at x = a.

(b) g (x) is differentiable on R.

(c) g (x) is continuous but not differentiable at x = b.

(d) g(x) is continuous and differentiable at either x = a or x = b but not both.

Answer 1

Correct option (a, c)

Explanation:

Since f(x) ≥ 1 ∀ x ∈ [a, b]

For g(x) L.H.D at x = a is zero.

and R.H.D at x = a

Thus, g(x) is not differentiable at x = a

Similarly, L.H.D at x = b is greater than 1.

So, g(x) is not differentiable at x = b.