Principles of Mathematics in Operations Research

280 Solutions

Problems of Chapter 12

12.1 Use the Mean Value Theorem: h : R >—» R is nondecreasing if h'(x) > 0.

y<x=> h(x) - h{y) = ti{c)(x - y) > 0 => /i(z) > /i(y).

.9%T) = xf'(x)x;fix\ f(x) = f(x) - /(0) = f'(c)x < xf'(x), 0<c<x.
So g'(x) > 0, V.T =4- g is nondecreasing.

12.2 Use the Mean Value Theorem: fi(y) - fi(x) = //(cj)(y - x). /' = 0 =»
/.; = 0, Vi; thus /,(y) = /',;(x') which means / is constant.

12.3 ||(0,0) = cos(0 + 2-0) = l, §£(0,0) = 2cos(0 + 2-0) =2;

0(0,0) = 0, 0(0,0) = 0, 0(0,0) = 0 and jg£(0,0) = 0.

f(x,y)=x + 2y + R 2 (x,y)(0,Q),

where jgf#(0,0) ^ 0 as {x,y) -* (0,0).

12.4
a) Let us take the first order Taylor's approximation for any nonzero direction
h,

f(x* + h) = f(x*) I Vf(x*)Th + R,(x*, h), R^f'h) - 0 as h - 6.

Since ,'y||i t} w |/i^7 V^2 /(£)/2, where £ = ,x* + a/j,, 0 < a < 1, we say that

/(.x* + /'0«/(x*) + V/(:r*)Tft.

Since x is a local minimize!', fix) < f(x + h), V/i small. Therefore, for
all feasible directions Vfix)Th > 0, where the left hand side is known as
the directional derivative of the function. Since we have an unconstrained
minimization problem, all directions h (and so are inverse directions —h) are
feasible,

Vfix*)Th > 0 > -Vfix*)Th = Vfix*)ri~h) > 0,V/i ^ 0.

Thus, we must have V/(.x*) = 9.

b) Let us take the second order Taylor's approximation for any nonzero (but
small in magnitude) direction h,

fix* + h)^ fix*) + Vfix*)Th -f l-hTV^2 fix*)h.

Since Vfix*) = 0, we have