Let f be any continuous function on [0, 2] and twice differentiable on (0, 2).

Question

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

Accepted Answer

f(0) = 0, f(1) = 1 and f(2) = 2

Let h(x) = f(x) - x

Clearly h(x) is continuous and twice differentiable on (0, 2)

Also, h(0) = h(1) = h(2) = 0

$$\therefore$$ h(x) satisfies all the condition of Rolle's theorem.

$$\therefore$$ there exist C1 $$\in(0, 1) such that h'(c1) = 0

\[\Rightarrow$$ f'(1) - 1 = 0 $$\Rightarrow$$ f'(c1) = 1

also there exist c2 $$\in(1, 2) such that h'(c2) = 0

\[\Rightarrow$$ f'(c2) = 1

Now, using Rolle's theorem on [c1, c2] for f'(x)

We have f''(c) = 0, c\[\in(c1, c2)

Hence, f''(x) = 0 for some x\[\in(0, 2).

Solutions · 1