Question

If $x+y\leq2,x \geq 0,y \geq 0$ the point at which maximum value of $3x + 2y$ attained will be

• A. $(0,0)$
• B. $(\frac {1}{2}, \frac {1}{2})$
• C. $(0, 2)$
• D. $(2,0)$

Answer 1

Answer: (D)

$(2,0)$

Answer 2

Given, $x+y \leq 2, x \geq 0$ and $y \geq 0$
Let $z=3 x+2 y$
Now, table for $x+y=2$

x
0
2
1

y
2
0
1

At $(0,0), 0+0 \leq 2$
$\Rightarrow 0 \leq 2$, which is true.
So, shaded portion is towards the origin.

$\therefore$ The corner points on shaded region are $O(0,0), A(2,0)$ and $B(0,2)$
At point $O(0,0), z=3(0)+2(0)=0$
At point $A(2,0), z=3(2)+2(0)=6$
At point $B(0,2), z=3(0)+2(2)=4$
Hence, maximum value of $z$ is 6 at point $(2,0)$.