For positive functions f,g, if g(x)=x * f(x), and g′(x) >= 0, g′′(x) <= 0, is f′(x) <= 0?

kaltEvallwsr

kaltEvallwsr

Answered question

2022-11-19

For positive functions f,g, if g ( x ) = x f ( x ), and g ( x ) 0, g ( x ) 0, is f ( x ) 0?

Answer & Explanation

Lillianna Salazar

Lillianna Salazar

Beginner2022-11-20Added 22 answers

Write f ( x ) = g ( x ) x and note that f ( x ) 0 if and only if h ( x ) := x g ( x ) g ( x ) 0.
Now observe that h ( 0 ) = 0. Therefore, it suffices to show that h is non-increasing, that is, h ( x ) 0. But we know that h ( x ) = x g ( x ) 0 if x 0.

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