ISI2015-MMA-79

Question

Let $g(x,y) = \text{max}\{12-x, 8-y\}$. Then the minimum value of $g(x,y)$ $ $ as $(x,y)$ varies over the line $x+y =10$ is

• A. $5$
• B. $7$
• C. $1$
• D. $3$

Answer 1

Answer $A$

Check if $12-x$ is maximum.

$\Rightarrow 12-x > 8 - y$

but $ y = 10 - x$ $\because x + y = 10\;\text{ (Given)}$

$\Rightarrow 12 - x > 8-(10-x)$

$\Rightarrow x < 7$

Now,

$g(x,\ y) = 12-x,\ \text {if }\ x < 7$

Now, the minimum value of $12-x$ is when $x$ has maximum value that is $7$

$\Rightarrow min(g(x,\;y)) = 12 - 7=5 \;\text{if} \;x < 7$

the minimum of $x-2$ is when x has minimum value $i.e.$ 7

$\Rightarrow min(g(x,\;y)) = 7-2 = 5 \;\text{if} \;x > 7$

$\therefore$ Answer$= 5$

$\therefore A$ is the correct answer.