retagged by
876 views
4 votes
4 votes

A superadditive function $f(\cdot)$ satisfies the following property $$f\left ( x_{1} +x_{2}\right )\geq f\left ( x_{1} \right ) + f\left ( x_{2} \right )$$

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

  1. $e^{x}$
  2. $\sqrt{x}$
  3. $1/x$
  4. $e^{-x}$
retagged by
Migrated from GO Electronics 2 years ago by Arjun

2 Answers

Best answer
3 votes
3 votes

Given that, a superadditive function $f(\cdot):$ $$f\left ( x_{1} +x_{2}\right )\geq f\left ( x_{1} \right ) + f\left ( x_{2} \right )$$

Since $x >1,$ a superadditive function can never be a decreasing function. So, optionc C and D can staright away be ruled out.

We can check options A and B by taking the value of $x_{1} = 2$ and $x_{2} = 3.$ 

  1. $f(x) = \sqrt{x}$
    $f(2+3) \geq f(2) + f(3)$
    $\implies f(5) \geq f(2) + f(3)$
    $\implies \sqrt{5} \geq \sqrt{2} + \sqrt{3}$
    $\implies 2.236 \geq 1.414 + 1.732$
    $\implies 2.236 \geq 3.146 \;{\color{Red} {\textbf{(False)}}}$

So, correct answer should be $(A).$

Reference:https://en.wikipedia.org/wiki/Superadditivity

selected by
0 votes
0 votes

Solve it using exploring the options

Let $X_{1}$ = 2 and $X_{2}$ = 3

A. f(x) = $e^{x}$

      $e^{5}$ > $e^{2}$ + $e^{3}$ ( using the calculator)

it satisfies the condition so option A is correct.

Answer:

Related questions

2 votes
2 votes
1 answer
1
go_editor asked Feb 13, 2020
952 views
It is quarter past three in your watch. The angle between the hour hand and the minute hand is ________.$0^{\circ}$$7.5^{\circ}$$15^{\circ}$$22.5^{\circ}$
3 votes
3 votes
2 answers
3
go_editor asked Feb 13, 2020
1,101 views
$a, b, c$ are real numbers. The quadratic equation $ax^{2}-bx+c=0$ has equal roots, which is $\beta$, then$\beta =b/a$$\beta^{2} =ac$$\beta^{3} =bc/\left ( 2a^{2} \right ...