A superadditive function f() satisfies the following property f(x 1+x 2) f(x 1)+f(x 2) Which of the following functions is a superadditive function for x>1?

GATE 2020 EC – Question 5

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

A superadditive function $f(\cdot)$ satisfies the following property

$$f(x_1+x_2)\ge f(x_1)+f(x_2)$$

Which of the following functions is a superadditive function for $x>1$?

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

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Correct answer: (A) $e^x$

Explanation

For $x_1,x_2>1$, $e^{x_1+x_2}=e^{x_1}e^{x_2}>e^{x_1}+e^{x_2}$, because the product of two numbers greater than $e$ exceeds their sum. $\sqrt x$ is concave (so $\sqrt{x_1+x_2}<\sqrt{x_1}+\sqrt{x_2}$), and $1/x$ and $e^{-x}$ are decreasing, so their values at the sum are smaller than either term.