Question

Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.

Options

• f is a constant function if f `(1/2) = 0`

• f is a constant function

• f is a constant function if f `(1/2) = "f"(3)`

• f is not a constant function

Answer 1

Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then f is a constant function if f `(1/2) = "f"(3)`

Explanation:

f '(x) = 0 ⇒ f (x) is constant in (0,1) and also in (2,4). But this does not mean that f(x) has the same value in both the intervals. However, if f(c) = f(d), where c ∈ ( 0,1) and d ∈ (2,4) then f(x) assumes the same value at all x ∈ (0,1) U (2, 4) and hence f is a constant function.