Problem 7. Let A € Rx, f: RR and bR". Define g: R R as g(x) f(Ax + b). = (a) If f is L-smooth (L>0), is g always smooth? If

you think g not smooth, please give your reason. If you think g is smooth, please find the smoothness parameter, i.e. a constant L₁ >0 such that ||Vg(x₁) - Vg(x2)|| ≤ L₁||X1 X2||- (7 marks) (b) Let hy be concave and h₂ be concave. Define h(x) = min{h₁(x), h₂(x)}. Is h also concave? Please justify your answer by either giving a proof or showing a counterex- ample. (8 marks) Hint: You can use the following inequality ||Ax||||A||||x, where ||A|| denotes the spectral norm of A.