Verify the Rolle’s theorem for each of the functions
f(x) = log (x2 + 2) – log3 in [– 1, 1].
Given: f(x) = log (x2 + 2) – log3
Now, we have to show that f(x) verify the Rolle’s Theorem
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:
f(x) = log (x2 + 2) – log3
Since, f(x) is a logarithmic function and logarithmic function is continuous for all values of x.
⇒ f(x) = log (x2 + 2) – log3 is continuous at x ∈ [-1,1]
Hence, condition 1 is satisfied.
Condition 2:
f(x) = log (x2 + 2) – log3
On differentiating above with respect to x, we get
⇒ f(x) is differentiable at [-1,1]
Hence, condition 2 is satisfied.
Condition 3:
∴f(-1) = f(1)
Hence, condition 3 is also satisfied.
Now, let us show that c ∈ (-1,1) such that f’(c) = 0
Put x = c in above equation, we get
∵, all the three conditions of Rolle’s theorem are satisfied
f’(c) = 0
⇒ 2c = 0
⇒ c = 0 ∈ (-1, 1)
Thus, Rolle’s theorem is verified.