INCREASING AND DECREASING FUNCTION 1 STRICTLY INCREASING FUNCTION

INCREASING AND DECREASING FUNCTION 1

STRICTLY INCREASING FUNCTION: � 2

� � � Thus, f(x) is strictly increasing on (a, b) if the values of f(x) decrease with increase in the value of x. Graphically, f(x) is decreasing on (a, b) if the graph y=f(x) moves down as x moves to the right. The graph in fig is the graph of strictly decreasing on (a, b). 4

� Interval (1, 2) f’(x) +ve -ve +ve f(x) Increasing Decreasing increasing 11

� Interval (-2, -1) f’(x) +ve -ve +ve f(x) Increasing Decreasing Increasing 12

TYPE II ON PROVING THE MONOTONICITY OF A FUNCTION ON GIVEN INTERVAL(S) � 14

� Ex 2: State when a function is said to be increasing function on [a, b]. Test whether the function f(x) = x 3 – 8 in increasing on [1, 2]. 15

Ex 3: Find the least value of a such that the function f given by f(x)=x 2+ax+1 is strictly increasing on (1, 2). � � � � � Given, f(x)=x 2+ax+1 f’(x)=2 x+a For f to be strictly increasing on(1, 2), f’(x)≥ 0 ��� (1, 2) Now 2 x+a ≥ 0 for all x∈(1, 2) Least value of 2 x+a ≥ 0 for all x∈(1, 2) 2× 1+a ≥ 0 a ≥− 2 Hence required value if x are given by -2≤��<∞ ∴���������� � 16

Exercise � � � Find the intervals in which the following function are increasing or decreasing. f(x) = 2 x 3 -15 x 2 + 36 x +1 f(x) = 2 x 3 -9 x 2 + 12 x - 5 20