Question : Which one of the following is the smallest?
Option 1: $\sqrt{101}-\sqrt{99}$
Option 2: $\sqrt{201}-\sqrt{199}$
Option 3: $\sqrt{301}-\sqrt{299}$
Option 4: $\sqrt{401}-\sqrt{399}$
Correct Answer: $\sqrt{401}-\sqrt{399}$
Solution :
The expressions given are differences in square roots of numbers that are 2 units apart.
For such expressions, the difference decreases as the numbers increase.
The function $f(x) = \sqrt{x}$ is concave up, meaning the rate of increase of $f(x)$ decreases as $x$ increases. Therefore, the difference decreases as numbers increase.
Therefore, among the given options, $\sqrt{401}-\sqrt{399}$ is the smallest.
Hence, the correct answer is $\sqrt{401}-\sqrt{399}$.
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