Question

Verify Rolle’s theorem for the following function:

f (x) = x² - 4x + 10 on [0, 4]

Answer 1

Since f (x) is a polynomial,

(i) It is continuous on [0, 4]

(ii) It is differentiable on (0, 4)

(iii) f (0) = 10, f (4) = 16 - 16 + 10 = 10

∴f(0) = f(4) = 10

Thus all the conditions on Rolle’s theorem are satisfied

The derivative of f (x) should vanish for at least one point c in (0, 4). To obtain the value of c, we
proceed as follows

f(x) = x² - 4x + 10

f'(x) = 2x - 4 = 2(x - 2)

∴ f'(x) = 0 ⇒ (x - 2) = 0

∴ x= 2

∴ ∃c = 2 in  (0,4)

We know that 2 ∈ (0, 4)

Thus Rolle’s theorem is verified.