Question

Let f : [2,4] → R be a differentiable function such that (x log_e x) f'(x) + (log_e x) f(x) + f(x) ≥ 1,

x ∈ [2,4] with f(2) = 1/2 and f(4) = 1/4.

Consider the following two statements:

(A): f(x) ≤ 1, for all ∈ [2,4]

(B): f(x) ≥ 1/8, for all ∈ [2,4]

Then, 

(1) Only statement (B) is true 

(2) Neither statement (A) nor statement (B) is true 

(3) Both the statement (A) and (B) are true 

(4) Only statement (A) is true

Answer 1

(3) Both the statement (A) and (B) are true 

Hence both A and B are true. 

LMVT on (yx (lnx)) not satisfied. 

Hence no such function exists. 

Therefore it should be bonus.