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Let f : [0, ∞) → [0,3] be a function defined by f(x) = { max {sin t : 0 ≤ t ≤ x}, 0 ≤ x ≤ π { 2 + cos x, x > π Then which of the following is true ?

1. f is continuous everywhere but not differentiable exactly at one point in (0,∞)
2. f is differentiable everywhere in (0,∞)
3. f is not continuous exactly at two points in (0,∞)
4. f is continuous everywhere but not differentiable exactly at two points in (0,∞)

Correct answer: f is differentiable everywhere in (0,∞)

Solution

The function f is defined piecewise, and both pieces are continuous and differentiable in their respective domains. Since the transition at x = π does not introduce any discontinuity or non-differentiability, f is differentiable everywhere in the interval (0, ∞).

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