When solving a minimax problem with a linear objective function by the method of Linear programming,

you should determine a feasible domain first.



Usually, in 2D problems (as in your problem with c = 4x + 5y), it is a polygon.



In the most simple version of the LP-method (which student usually learn first), the algorithm is to calculate the values

of the objective function at the corners of the feasibility polygon, and then to select that vertex (corner),

where the objective function has minimal / (maximal) value.


To calculate the value of the objective function  c = 4x + 5y at a vertex, simply substitute the coordinate values 
of the vertex into the function's expression.