154 MATHEMATICS Proof a Let x x a b be such that x x Then...
50.1K
Verified Solution
Link Copied!
Question
Algebra
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
Answer & Explanation
Solved by verified expert
Get Answers to Unlimited Questions
Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!
Membership Benefits:
Unlimited Question Access with detailed Answers
Zin AI - 3 Million Words
10 Dall-E 3 Images
20 Plot Generations
Conversation with Dialogue Memory
No Ads, Ever!
Access to Our Best AI Platform: Zin AI - Your personal assistant for all your inquiries!
