Increasing/Decreasing composites functions

[jordanfc]

Given that u(x) is always positive and u'(x) < 0, I need to find values of x so that f(x) and g(x) are increasing. f(x) = [u(x)]^2 and g(x) = u(u(x)).

for f(x) is increasing when f'(x) > 0. so f'(x) = 2u(x) => 2u(x) > 0. would f(x) always be increasing since 2u(x) will always be increasing ( u(x) is always positive, 2u(x) will always be positive as well)?

and for g'(x) > 0 => [u(u(x))]' => u'(u(x)*(u(x))' = 2u'(u(x))...set larger than 0...since u'(x) is always negative and 2u'(u(x)) is bounded by 2u', does that inequality not hold and so g(x) is never increasing?

 

[EnumaElish]

f'(x) is not 2u(x), you need to apply the chain rule. You must also use the chain rule when you analyze g'(x).