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Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Maximum and minimum element.

A set S is said to have a minimum if there is an element a in S such that \(a \leq y\) for all y in S. Similarly, S is said to have a maximum if there is an element b in S such that \(b \geq y\) for all y in S. If S=\((y:y=\frac{2x+3}{x+2}, x \geq 0)\), which one of the following statements is correct?

- S has both a maximum and a minimum
- S has a minimum but no maximum
- S has a maximum but no minimum
- S has neither a maximum nor a minimum

Equation

Roots

Algebra

But try the problem first...

Answer:S has a minimum but no maximum

Source

Suggested Reading

B.Stat Objective Problem 715

Challenges and Thrills of Pre-College Mathematics by University Press

First hint

y=f(x)=\(\frac{2x+3}{x+2}\)

or, \(f'(x)=\frac{1}{(x+2)^{2}}>0\)

Second Hint

So its a strictly increasing function

So it attains its minimum at x=0

Final Step

As given that function is defined on [0, infinity)

or, S has a minimum but no maximum.

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