GATE 2023 ME – Question 48
The smallest perimeter that a rectangle with area of 4 square units can have is ______ units.
(Answer in integer)
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Correct answer: 8
Explanation
For sides $a$ and $b$ with $ab = 4$, the perimeter is $2(a + b) = 2\left(a + \frac{4}{a}\right)$. Setting the derivative to zero gives $1 - \frac{4}{a^2} = 0$, so $a = 2$ and $b = 2$ (a square). The smallest perimeter is $2 \times 4 = 8$ units. (By the AM-GM inequality, $a + b \ge 2\sqrt{ab} = 4$.)