inequalities with maximums and minimums!

Aug 2010
2
0
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!? (Thinking)
 
Jan 2010
278
138
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:
\(\displaystyle \begin{aligned}
y &\le -x + 9 \\
y &\le 3x \\
x &\ge 0 \\
y &\ge 0
\end{aligned}\)
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
\(\displaystyle f(x) = 10x + 11y\).

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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