"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"

Question

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).

"

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