141 Exercises 5.5.1 (a) Let I be an interval in Rand let f,g: I →Rbe convex functions. Prove that c f , f +g, and max{f,g}are convex functions onI, where c ≥0 is a constant. (b) Find two convex functions f andgon an interval I inRsuch that f · gis not convex. 5.5.2 Let f : R→Rbe a convex function. Givena,b∈R, prove that the function defined by g(x) =f (ax+b), for x ∈R is also a convex function on R. 5.5.3 ▶Let I be an interval and let f : I →Rbe a convex function. Suppose that φ is a convex, increasing function on an interval J that contains f (I). Prove that φ◦ f is convex onI. 5.5.4 ▷Prove that each of the following functions is convex on the given domain: (a) f (x) =ebx,x ∈R, where bis a constant. (b) f (x) =xk, x ∈[0,∞) andk ≥1 is a constant. (c) f (x) =−ln(1−x), x ∈(−∞,1). (d) f (x) =−ln ex 1+ex , x ∈R. (e) f (x) =xsinx, x ∈(−π 4, π 4). 5.5.5 ▷Prove the following: (a) If a,bare nonnegative real numbers, then a+b 2 ≥ √ab. (b) If a1,a2, . . . ,an, where n≥2, are nonnegative real numbers, then a1 +a2 +· · ·+an n ≥ (a1 · a2· · ·an) 1/n. 5.6 Nondifferentiable Convex Functions and Subdifferentials In this section, we introduce a generalized differentiation concept that is useful in the study of optimization problems in which the objective functions may fail to be differentiable. Definition 5.6.1 Let f : R→Rbe a convex function. A number u∈Ris called a subderivative of the function f at x0 if u· (x−x0) ≤f (x)−f (x0) for all x ∈R. (5.8) The set of all subderivatives of f at x0 is called the subdifferential of f at x0 and is denoted by ∂ f (x0).
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