Introduction to Mathematical Analysis I - Second Edition

141 4.7.4 Let f : R → R be a convex function. Fix a , b ∈ R and define the function g by g ( x ) = f ( ax + b ) , for x ∈ R Prove that ∂ g ( ¯ x ) = a ∂ f ( a ¯ x + b ) . 4.7.5 B Let f : R → R be a convex function. Suppose that ∂ f ( x ) ⊂ [ 0 , ∞ ) for all x ∈ R . Prove that f is monotone increasing on R .