Let f: R - R and g: R - R be two non-constant differentiable functions satisfying f'(x) = e^((f(x) - g(x))) g'(x) for all real x, with f(1) = g(2) = 1. Which statement(s) is/are TRUE?

Question

Let f: R -> R and g: R -> R be two non-constant differentiable functions satisfying f'(x) = e^((f(x) - g(x))) g'(x) for all real x, with f(1) = g(2) = 1. Which statement(s) is/are TRUE?

Accepted Answer

g(1) > 1 - ln 2. From e^(-f) f' = e^(-g) g', integrating yields -e^(-f(x)) = -e^(-g(x)) + C. Using f(1)=g(2)=1 and the resulting inequalities gives f(2) > 1 - ln 2 and g(1) > 1 - ln 2 (this is the classic JEE Advanced 2018 result).

Let f: R -> R and g: R -> R be two non-constant differentiable functions satisfying f'(x) = e^((f(x) - g(x))) g'(x) for all real x, with f(1) = g(2) = 1. Which statement(s) is/are TRUE?

Solution

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