Principles of Mathematics in Operations Research

170 12 Differentiation

Remark 12.1.4 The converse is not true. One can construct continuous

functions which fail to be differentiate at isolated points.

Let us state some properties: Suppose / and g are denned on [a, b] and are

differentiable at x G [a,b]. Then, f + g, f • g and f/g are differentiable at x,

and

(a) (f + gy(x) = f'(x)+g'(x).

(b) (f-gy(x) = f'(x)g(x) + f(x)g>(x).

(c) (f/g)'(x) = /'(*)g(*)-/(«)g'(«), g{x) + o.

(d) Chain Rule: If h(t) = (g°f)(t) = g(f{t)), a < t < b, and if/ is continuous
at [a, b], f exists at x £ [a,b], g is defined over range of / and g is
differentiable at f(x). Then, h is differentiable at x and

h'(x) = g'(f(x))f(x).

Example 12.1.5 (Property (c)) The derivative of a constant is zero. If

f(x) — x then f'(x) — 1. // f(x) — x • x = x^2 then f'(x) — x + x = 2x by

property (b). In general, if f(x) = xn then f'(x) = nxn~^1 , n € N. If f{x) =

i = x~l then f'(x) = ^ = -x~^2. In this case, x ^ 0. if f(x) = x~n, n E N

then f'(x) = — nx~ -("+^1 ). Thus, every polynomial is differentiable, and every

rational function is differentiable except at the points where denominator is

zero.

Example 12.1.6 (Property (d)) Let

/(*){

X sin -, x ^ 0

0, x = 0

Then, f'(x) = sin \ - \ cos A, x ^ 0. At x = 0, A is no* de/med /(t|l^0) =

sin j. yls £ —>• 0, £/ie Kraii does not exist, thus /'(0) does not exist.

12.2 Mean Value Theorems

Definition 12.2.1 Let f : [a,b] H-» K. We say f has a local maximum at
peXif35>0 B f{q) < f(p), V</ G X with d{p,q) < 5. Local minimum is
defined similarly.

Theorem 12.2.2 Let f : [a,b] i-> K. /// has a local maximum (minimum)
at x £ (a,b) and if f'(x) exists, then f'(x) = 0.

Proof. We will prove the maximum case:
Choose S as in the definition: a < x — 5<x<x + 5<b.
lix-8<t<x, then mtZfJx) > 0. Let t -> x => /'(a:) > 0.

If x < t < x + (5, then f{t)tZfx{x) < 0. Let t -> oo => /'(a;) < 0.
Thus, /'(x) = 0. D