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**Zexuo** This one also comes from Purcell and Varberg, fifth edition, chapter 4, section 2 problem 41b.

Given f'(x) > 0 and g'(x) > 0, what simple additional conditions would guarantee that f(x)g(x) increases for all x?

The answer key has f(x) > 0 and g(x) > 0, but I have a counter example: f(x) = x and g(x) = x^2 + 1. f(x) < 0 for x < 0

but the derivative of f(x)g(x) > 0, and therefore f(x)g(x) increases for all x.

I got only the following condition from the product rule: f(x)/f'(x) > -g(x)/g'(x).