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Math Help - inequalities with maximums and minimums!

  1. #1
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    inequalities with maximums and minimums!

    i dont understand how to find the maximum/minimum of these two equations, help please!

    Given the constraints
    x+y is less than or equal to 9
    3x-y is greater than or equal to 0
    x is greater than or equal to 0
    y is greater than or equal to 0
    the minimum value for the function f(x)=10x + 11y is..?

    Given the constraints
    3x+4y is greater than or equal to 60
    x+8y is greater than or equal to 40
    11x+28y is less than or equal to 380
    x is greater than or equal to 0
    y is greater than or equal to 0
    the maximum value of the function f(x)=15x+25y occurs at the corner point..?

    explanations please!?
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  2. #2
    Senior Member eumyang's Avatar
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    Given the constraints
    x+y is less than or equal to 9
    3x-y is greater than or equal to 0
    x is greater than or equal to 0
    y is greater than or equal to 0
    Graph the four inequalities. (I'm assuming you know how to do this.) It may help to change them into slope-intercept form, if you can:
    \begin{aligned}<br />
y &\le -x + 9 \\<br />
y &\le 3x \\<br />
x &\ge 0 \\<br />
y &\ge 0<br />
\end{aligned}
    You'll find that the intersection of these graphs (all of these graphs are solid lines where one area that each line divides gets shaded) will be a triangle. The vertices of the intersecting area are your critical points that you will need to test into the function
    f(x) = 10x + 11y.

    In our case, there are three critical points. Two of them are easy to find. The other is a point that is an intersection of two lines. Plug in these points into f(x). The smallest value of f(x) is your minimum.
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