Suppose U and g are two twice differentiable functions of x, both of them increasing and concave, with U’ >= 0, U” <= 0, g’ >= 0, and g” <= 0. Prove that the composite function f(x) = g(U(x)) is also increasing and concave.

Chaim Ferguson

Chaim Ferguson

Answered question

2022-10-25

Suppose U and g are two twice differentiable functions of x, both of them increasing and concave, with U 0 , U 0 , g 0, and g 0. Prove that the composite function f ( x ) = g ( U ( x ) ) is also increasing and concave.

Answer & Explanation

Bridget Acevedo

Bridget Acevedo

Beginner2022-10-26Added 19 answers

Hint: try using the chain rule: f ( x ) = U ( x ) g ( U ( x ) )
Danika Mckay

Danika Mckay

Beginner2022-10-27Added 5 answers

Use the chain rule on the sets where it has sense : you will have that
f = U g ( U ) 0
by hypothesis, and
f = U g ( U ) + ( U ) 2 g ( U ) 0 ,
giving you that f is increasing and concave.

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