Question

Verify Rolle's theorem of the function log (x² + 2) – log 3 on [–1, 1]

Answer 1

Let f(x) = log(x² + 2) – log3 

Clearly f is continous on [–1, 1] and 

f is derivable on (–1, 1)

Also, f(–1) = log (1 + 2) – log 3 = log 3 – log 3 = 0 

f(1) = log (1 + 2) – log 3 = log 3 – log 3 = 0 

∴ f(–1) = f(1) 

f satisfies all the conditions of Rolle's theorem. 

∴ There exists C ∈ (–1, 1) such that f¹ (c) = 0

But f(x) = log (x² + 2) – log 3

2c = 0

⇒ c = 0 ∈ (–1, 1) 

Hence Rolle's theorem is verified.