Let h be a convex decreasing function and g a convex function. Is it true that h(g(x)) is a quasi-concave function?

Eliza Gregory

Eliza Gregory

Answered question

2022-10-25

Let h be a convex decreasing function and g a convex function. Is it true that h ( g ( x ) ) is a quasi-concave function?

Answer & Explanation

periasemdy

periasemdy

Beginner2022-10-26Added 15 answers

Yes.Since g is convex, for λ [ 0 , 1 ] ,,
g ( λ x + ( 1 λ ) y ) λ g ( x ) + ( 1 λ ) g ( y ) .
and
λ g ( x ) + ( 1 λ ) g ( y ) max [ g ( x ) , g ( y ) ]
Since h is decreasing,
h [ g ( λ x + ( 1 λ ) y ) ] h [ λ g ( x ) + ( 1 λ ) g ( y ) ] h ( max [ g ( x ) , g ( y ) ] ) min [ h ( g ( x ) ) , h ( g ( y ) ) ] .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?