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Question:

Let f:R→R and g:R→R be two non-constant differentiable functions. If f'(x)=(e^(f(x)−g(x)))g'(x) for all x∈R, and f(1)=g(2)=1, then which of the following statement(s) is (are) TRUE?

f(2)<1−loge2

g(1)>1−loge2

f(2)>1−loge2

g(1)<1−loge2

Solution:

f'(x)=e^(f(x)−g(x))g'(x) ∀x∈R ⇒e^−f(x)f'(x)−e^−g(x)g'(x)=0 ⇒∫(e^−f(x)f'(x)−e^−g(x)g'(x))dx=C ⇒−e^−f(x)+e^−g(x)=C ⇒−e^−f(1)+e^−g(1)=−e^−f(2)+e^−g(2) ⇒−e^−1+e^−g(1)=−e^−f(2)+e^−1 ⇒e^−f(2)+e^−g(1)=2e^−1 ⇔e^−f(2)<2e^−1 and e^−g(1)<2e^−1 ⇒−f(2)<loge2 and −g(1)<loge2 ⇒f(2)>1−loge2 and g(1)>1−loge2.