APPLICATION OF DERIVATIVES |
Introduction
Increasing function: (a) Strictly increasing function: A function f(x) is c/d strictly increasing function in its domain if x1 < x2  For strictly increasing function f'(x) > 0  domain Dumb Question: How f'(x) > 0n domain for strictly increasin function. Ans: As x1 < x2  f(x1) < f(x2) So, f(x) < f(n + h) f'(x) =  f'(x) > 0 Types of strictly increasing function: (1) Concave up: When f'(x) > 0 & f"(x) > 0 domain  (2) Concave down: f'(x) > 0 & f"(x) < 0 domain  (3) When f'(x) > 0 & f"(x) = 0 domain f'(x)> 0 & f"(x) = 0  Increasing function: A function f(x) is said to be non decreasing if for x1 < x2  Laet us see in fig.  For portion ABCD, x1 < x2 f(x1) < f(x2)  for BC, x1 < x2 f(x1) = f(x2) Dumb Question: What is diff. b/w strictly increasing & increasing function ? Ans: Strictly function for x1 < x2, f(x2) is always greater than f(x1) but in increasing function for x1 < x2, f(x2 may be greater or equal to f(x1). Decreasing functions: (a) Strictly decreasing function:  A function f(x) ic c/d strictly decreasimg in its domain if x1 < x2 f(x1) > f(x2) For strictly decreasing function. f'(x) < 0 Dumb Question: How f'(x) < 0 for strictly decreasing ? Ans: As x1 < x2 f(x1) > f(x2) So, f(x + h) < f(x)  f'(x) < 0 Types of strictly decreasing function: (i) Concave up:  When f'(x) < 0 f''(x) > 0 domain (ii) Concave down:  When f'(x) < 0 & f''(x) < 0 domain (iii)  When f'(x) < 0 & f'(x) = 0 domain (5) Non-increasing function:  A function f(x) is c/d non-increasing if for x1 < x2 for x1 < x2 f(x1)  f(x2) For AB & CD, x1 < x2 f(x1 > f(x2) Bc, x1 < x2 f(x1 = f(x2)Illustration: Find interval in which f(x) = x3 - 3x2 - 9x + 20 is strictly increasing or decreasing. Ans: f(x) = x3 - 3x2 - 9x + 20 f'(x) = 3x2 - 6x - 9 f'(x) = 3(x - 3)(x + 1) For strictly increasing f'(x) > 0  3(x - 3)(x + 1) > 0 (x - 3)(x + 1) > 0 (x + 1) < 0 or (x - 3) > 0 x < - 1 or x > 3  x (-  , - 1) U (3, ) For strictly decreasing f'(x) < 0 (x + 1)(x - 3) < 0 x (- 1, 3) |
