Try this beautiful problem based on Real valued function, useful for ISI B.Stat Entrance

Real valued functions | ISI B.Stat TOMATO 690


Let \(f(x)\) be a real-valued function defined for all real numbers x such that \(|f(x) – f(y)|≤(1/2)|x – y|\) for all x, y. Then the number of points of intersection of the graph of \(y = f(x)\) and the line \(y = x\) is

  • 0
  • 1
  • 2
  • none of these

Key Concepts


Limit

Calculas

Real valued function

Check the Answer


But try the problem first…

Answer: \(1\)

Source
Suggested Reading

TOMATO, Problem 690

Challenges and Thrills in Pre College Mathematics

Try with Hints


First hint

Now,

\(|f(x) – f(y)| ≤ (1/2)|x – y|\)


\(\Rightarrow lim |{f(x) – f(y)}/(x – y)|\)( as x -> y ≤ lim (1/2)) as x – > y


\(\Rightarrow |f‟(y)| ≤ ½\)


\(\Rightarrow -1/2 ≤ f‟(y) ≤1/2\)


\(\Rightarrow -y/2 ≤ f(y) ≤ y/2\) (integrating)


\(\Rightarrow -x/2 ≤ f(x) ≤ x/2\)

Can you now finish the problem ……….

Second Hint

Real valued function graph

Therefore from the picture we can say that intersection point is \(1\)

Subscribe to Cheenta at Youtube