Question

f(x) = log(x² + 2) – log3 in [–1, 1]

Answer 1

We have, f(x) = log(x² + 2) – log3

We know that x² + 2 and logarithmic function are continuous and differentiable

∴ f(x) = log(x² + 2) – log3 is also continuous and differentiable.

Now f(–1) = f(1) = log3 - log3 = 0

So, conditions of Rolle's theorem are satisfied.

Hence, there exists atleast one c ∈ (–1, 1) such that f'(c) = 0

f(x) = `(2"c")/("c"^2 + 2) - 0` = 0

⇒ c = 0 ∈ (–1, 1)

Hence, Rolle's theorem has been verified.