A superadditive function satisfies f(x 1+x 2) f(x 1)+f(x 2). Which function has this property for x₁,x₂>1?

GATE 2020 AG – Question 5

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

A superadditive function satisfies $f(x_1+x_2)\ge f(x_1)+f(x_2)$. Which function has this property for x₁,x₂>1?

$e^x$
$\sqrt x$
$1/x$
$e^{-x}$

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Show answer and explanation

Correct answer: (A) $e^x$

Explanation

For f=eˣ, divide the required inequality by e^(x₁+x₂): it becomes 1≥e^(−x₁)+e^(−x₂). Each term is less than 1/e, so their sum is less than 2/e<1. Thus A works. The square root and the two decreasing functions fail the stated inequality for positive arguments.

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