Question

maximize z=5x+2y
Subject to 5x-y <15
2x + y >10
x > 3
y < 8
name the maximum value and the points

Answer 1

Input interpretation:

plot $2x + y < 10\land 5x - y < 15\land x < 3\land y < 8$

Inequality plot:

Let's evaluate partial derivatives of $5x + 2y$

$$\frac {\partial f}{\partial x} = 5; \frac {\partial f}{\partial y} = 2$$

Consequently this function has no critical points inside the domain on the plot $\Rightarrow$ it has critical points on the bound of domain. The curves that draw this domain are regular $\Rightarrow$ the critical points can be only the points of intersection of curves.

These points are:

(3;0)

(3;4)

(1;8)

Substitution this values to f gives us that maximum value is achieved in the point (3;4) and equals 23.