Question:

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

Updated On: May 19, 2024
  • $(0,0)$
  • $(\frac {1}{2}, \frac {1}{2})$
  • $(0, 2)$
  • $(2,0)$
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The Correct Option is D

Solution and Explanation

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 $(2,0)$.
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Concepts Used:

Linear Programming

Linear programming is a mathematical technique for increasing the efficiency and effectiveness of operations under specific constraints. The main determination of linear programming is to optimize or minimize a numerical value. It is built of linear functions with linear equations or inequalities restricting variables.

Characteristics of Linear Programming:

  • Decision Variables: This is the first step that will determine the output. It provides the final solution to the problem.
  • Constraints: The mathematical form in which drawbacks are expressed, regarding the resource.
  • Data: They are placeholders for known numbers to make writing complex models simple. They are constituted by upper-case letters.
  • Objective Functions: Mathematically, the objective function should be quantitatively defined.
  • Linearity: The function's relation between two or more variables must be straight. It indicates that the variable's degree is one.
  • Finiteness: Input and output numbers must be finite and infinite. The best solution is not possible if the function consists infinite components.
  • Non-negativity: The value of the variable should be either positive (+ve) or 0. It can't be a negative (-ve) number.