Maximize the function

f(x, y) = 6x + 7y in the region determined by the following constraints.

3x + 2y≤ 18

3x + 4y≥ 12

x≥0

y≥0

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• z = 6x + 7y would define a plane in 3-space, so the maximum value will occur somewhere on the boundary of the region. But all the boundaries of the region are straight lines, so the maximum value will occur at one of the vertices.

The vertices of the region are (0,3), (0,9), (4,0), (6,0). The corresponding values of f(x,y) are

21, 63, 24, 36. The largest is 63.

And that's the answer, 63. Achieved when x = 0 and y = 9.

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• check the 4 corners (0,3) , (0,9) , ( 4 , 0 ) ,( 6 , 0 )

f(0,9)=63

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• TedS is right. Always check the corners.

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• You thought of graphing/shading? Ultimately, you'll end up drawing a quadrilateral with the constraints. Then, just draw (f, x) and make note of the maixmum value it obtains. If it's none obvious, just make note of which side (constraint equation) it intersects with. Then, if need be, you can solve that system of two linear equations of two unknowns.

I don't know which is quicker, drawing + 1 potential equation, vs finding all for points like in ted s's answer.

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