A question on an inequality relating a function and its derivative

manpreet Best Answer 3 years ago

Let f:[0,1]→Rf:[0,1]→R be a differentiable function such that f(0)=0f(0)=0 and |f′(x)|≤k|f(x)|∀x∈[0,1]|f′(x)|≤k|f(x)|∀x∈[0,1],(k>0k>0), then which of the following is always true?

(A) f(x)=0∀x∈Rf(x)=0∀x∈R
(B) f(x)=0∀x∈[0,1]f(x)=0∀x∈[0,1]
(C) f(x)≠0∀x∈[0,1]f(x)≠0∀x∈[0,1]
(D) f(1)=kf(1)=k

This question appeared in a test I gave today (its obviously completed). I would love a hint on how to approach this question, and also some insight on how I should ha