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.

Question

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.

Bridget Acevedo · Accepted Answer

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

Danika Mckay · Accepted Answer

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

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.

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