154 MATHEMATICS Proof a Let x x a b be such that x x Then by

Last updated: 10/10/2023

154 MATHEMATICS Proof a Let x x a b be such that x x Then by Mean Value Theorem Theorem 8 in Chapter 5 there exists a point c between and X x such that i e i e Thus we have Rationalised 2023 24 f x f x f c x x f x f x 0 f x f x X x Hence f is an increasing function in a b f x f x for all x x a b is increasing on R Solution Note that The proofs of part b and c are similar It is left as an exercise to the Remarks There is a more generalised theorem which states that if f x for x in an interval excluding the end points and fis continuous in the interval then fis increasing Similarly if f x 0 for x in an interval excluding the end points and fis continuous in the interval then fis decreasing Example 8 Show that the function f given by f x 3x 6x 4 as f c 0 given 3 x 2x 1 1 t to be po hod 3 x 1 1 0 in every interval of R Therefore the function fis increasing on R Example 9 Prove that the function given by f x cos x is a decreasing in 0 b increasing in 1 2 and c neither increasing nor decreasing in 0 2