Given the optimization problem:

(x,y)R2maxf(x,y)=5x24y2+45x+30y

under constraint :2x+4y=12

of which the contour lines of the function f are illustrated in the given figure (see picture attached).

--> Indicate on the graph the point of coordinates (x,y) corresponding to the optimal solution to this problem.

--> What is the value of the Lagrange multiplier associated with this optimal solution?

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I'm not sure I understand how the maximum point of f(x,y) under the constraint can be found just by looking at the graph. Is there a method to do this or do I still have to calculate it with the gradient method or another method?

I would really appreciate a step-by-step solution if that's the case.