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AP EAMCET202022 Sep 2020Evening ShiftMathematicsFunctionsActual

Let f: R R be a continuous function such that for any two real numbers x and y , |f(x)-f(y) 10| x-. .y |²⁰¹ , then

Options

  1. Af(2019)=f(2020)+1
  2. Bf(2019)+f(2022)=2 f(2021)
  3. Cf(2019)=f(2020)+8
  4. Df(2019)=f(2020)+2

Correct answer

B. f(2019)+f(2022)=2 f(2021)

Step-by-step solution

Given, f: R R is continuous aligned & |f(x)-f(y)| 10|x-y|²⁰¹ & |f(x)-f(y)| |x-y| 10|x-y|²⁰⁰ aligned Put, y=x+h and taking limit h 0 on both sides aligned & _ h 0 |f(x)-f(x+h)| |x-(x+h)| _ h 0 |x-(x+h)|²⁰⁰ & _ h 0 | f(x+h)-f(x) h | 0 & |f^ (x) | 0 f^ (x)=0 & f(x)= constant & f(2019)= constant & f(2021)=f(2022)= constant & aligned So, f(2019)+f(2022)=2 f(2021) boths are constant Hence, option (2) is correct.

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