Question

The objective function of LPP defined over the convex set attains its optimum value at
1) At least two of the corner points
2) All the corner points
3) At least one of the corner points
4) None of the corner points

Answer 1

Hint:
The objective function is a function in the LPP which is to be optimized. The LPP objective function either has maximum value or minimum value or has no solution. A convex set is a region such that for every pair of points within the region, every point on the line segment must be within the region. In the problem, we need to find at which point the optimum value is attained within the region. So, we need to find the feasible region and check the corner points of the feasible region at which the function attains its optimum value.

Complete step by step solution:
First, consider the objective function,
Let the objective function be \[Z = ax + by\]
Now, we need to find the point at which the objective function attains its optimum value over the convex set.
So, the objective function \[Z\] has optimum value either the value is maximum or minimum.
As the variables, \[x\] and \[y\] are subject to constraints which define the linear inequalities.
Hence, by solving the inequalities we will find the feasible region which will have the corner points and the optimum value at which the objective function attains its optimization.
Also, the optimum value occurs at atleast one of the corner points of the feasible region.
Hence, the objective function over the convex set attains its optimum value at atleast one of the corner points.

Thus, option C is correct.

Note:
While solving the LPP problem, the objective function can attain its optimum value and have two options, one of getting maximum value and the other is of getting minimum value. When the objective function does not attain its optimum value implies that the objective function has no solution.