Step 1:

The corner points are $(0,2),(3,0),(6,0)$ and $(0,5)$

The objective function $F=4x+6y$

At the points $(x,y)$ the objective function subject to $F=4x+6y$

Step 2:

At $(0,2)$ the objective function $F=4x+6y\Rightarrow 4\times 0+6\times 2=12$

At $(3,0)$ the objective function $F=4x+6y\Rightarrow 4\times 3+6\times 0=12$

At $(6,0)$ the objective function $F=4x+6y\Rightarrow 4\times 6+6\times 0=24$

At $(6,8)$ the objective function $F=4x+6y\Rightarrow 4\times 6+6\times 8=72$

At $(0,5)$ the objective function $F=4x+6y\Rightarrow 4\times 0+6\times 5=30$

Step 3:

The minimum value of the objective function is 12 and this occurs at both the point $(0,2)$ and $(3,0)$

Hence the minimum value of $F$ occurs at any point on the line segment joining the points $(0,2)$ and $(3,0)$

The correct option is $D$