Mathematics - Area Under Curves Question with Solution | TestHub

The objective function to be maximized is $Z=x+y$.

The given constraints are:
$x+y\le 10$
$5x+3y\ge 15$
$x\le 6$
$x\ge 0,y\ge 0$

We find the vertices of the feasible region by solving the boundary equations of the constraints. The vertices are:
$(0,5)$ from the intersection of $x=0$ and $5x+3y=15$
$(0,10)$ from the intersection of $x=0$ and $x+y=10$
$(6,4)$ from the intersection of $x=6$ and $x+y=10$
$(6,0)$ from the intersection of $x=6$ and $y=0$
$(3,0)$ from the intersection of $y=0$ and $5x+3y=15$

Evaluating the objective function $Z=x+y$ at each vertex:
At $(0,5)$, $Z=0+5=5$
At $(0,10)$, $Z=0+10=10$
At $(6,4)$, $Z=6+4=10$
At $(6,0)$, $Z=6+0=6$
At $(3,0)$, $Z=3+0=3$

The maximum value of $Z$ is $10$, which occurs at two distinct vertices: $(0,10)$ and $(6,4)$.

Since the maximum value occurs at two distinct vertices, it will also occur at every point on the line segment joining these two vertices (which lies on the line $x+y=10$). Because a line segment contains an infinite number of points, the maximum value occurs at infinitely many points.

Answer: occurs at infinitely many points