Consider the function f : IR3 → IR defined byf (x) = 12 (x1)2 + 12 (x2)2 + 12 (x3)2 + ln(1 + (3x1 − x2 + 7x3 − 2)2) + 1.1(5x2 + 2x3 − 1)2(a) (5 points) Let h : IR2 → IR be defined by h(y) = ln(1 + (y1)2) + 1.1(y2)2. Show that∥∇2h(y)∥2 ≤ 2.2 for all y ∈ IR2.(b) (20 points) Suppose that the steepest descent with constant stepsize α = 0.01466 is applied tominimize f . Argue that any accumulation point of the sequence generated is a stationary pointof f

Consider the function f : IR3 → IR defined byf (x) = (x1)2 + (x2)2 + (x3)2 + ln(1 + (x1 + x2 − x3 − 7)4) + ln(1 + (3x1 + x3 + 2)4)(a) (5 points) Let h : IR2 → IR be defined by h(y) = ln(1 + (y1)4) + ln(1 + (y2)4). Show that∥∇2h(y)∥2 ≤ 6 for all y ∈ IR2.

5 points) Let h : IR2 → IR be defined by h(y) = ln(1 + (y1)2) + 1.1(y2)2. Show that∥∇2h(y)∥2 ≤ 2.2 for all y ∈ IR2

Suppose that the function h is defined, for all real numbers, as follows.=hx −14x2    ≤if x−2−+x12    <−if 2≤x2−3    >if x2Find h−2, h1, and h5.

Suppose that the function g is defined, for all real numbers, as follows.=gx 2    ≤if x−2−+x122    <−if 2<x2+−14x1    ≥if x2Find g−5, g−2, and g−1.

Let f, g : R → R be differentiable on [1, 2]. Suppose that f (1) = g(1) − 1 andf ′(x) ≤ g′(x) − 1 for all x ∈ [1, 2].Prove thatg(2) ≥ f (2) + 2.Hint) Consider the function h(x) := g(x) − x − f (x). Prove that h is non-negative on [1, 2