State the Convex polygon theorem.

Find convex region from the following figures.

The corner points of the feasible region determined by the following system of linear inequalities:
$2x+y\le 10,x+3y\le 15,x,y\ge 0$ are $\left(0,0\right),\left(5,0\right),\left(3,4\right)$ and $\left(0,5\right)$
Let $Z=sx+ty,$ where $s,t>0.$
Condition on $s$ and $t$ so that the maximum of $Z$ occurs at both $\left(3,4\right)$ and $\left(0,5\right)$ is

The feasible region of an $\mathrm{LPP}$ is shown in the figure. If $z=3x+9y$, then the minimum value of $z$ occurs at 

The shaded region is the solution set of the inequalities 

The maximum value of $z=6x+8y$ subject to $x-y\ge 0, x+3y\le 12, x\ge 0, y\ge 0$ is

Solve graphically the following linear programming problem:

Maximize or minimize $Z=x+2y$ subject to constraints $x+2y\ge 100, 2x-y\le 0, 2x+y\le 200 \& x\ge 0, y\ge 0$.

A furniture trader deals in only two items - chairs and tables. He has $50,000$ rupees to invest and a space to store at most $35$ items. A chair costs him $1000$ rupees and a table costs him $2000$ rupees . The trader earns a profit of $150$ rupees and $250$ rupees on a chair and table, respectively. Formulate the above problem as an LPP to maximise the profit and solve it graphically.