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Find the maximum

  1. Nov 8, 2015 #1
    • Member warned about posting without the homework template
    Hello

    This is what I have attempted so far. But now I'm at utter loss at how to calculate the rest Can you help? Thanks in advance.
     

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  3. Nov 8, 2015 #2

    Mark44

    Staff: Mentor

    Your function is defined on a closed, bounded region. Check the four boundaries for the maximum value.
     
  4. Nov 8, 2015 #3
    What does that mean? Could you explain more explicitly?
     
  5. Nov 8, 2015 #4

    Mark44

    Staff: Mentor

    Your function is defined on the square [0, 2] X [0, 2]. Along each of the four sides of this square your function simplifies to a single-variable function. For example, on the lower edge of the square, y = 0 and x varies from 0 to 2. So f(x, y) = f(x, 0). This is a function of x alone. Any maximum value will occur where the derivative is zero or at an endpoint of this edge.

    Do something similar for each of the four edges.
     
  6. Nov 8, 2015 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You say that (1/4,1) is a local minimum. It is a lot more than that: it is the global minimum in the entire plane ##\mathbb{R}^2##; and because the point (1/4,1) is feasible (satisfies all the constraints) it is the overall minimum in your constrained problem. Because (1/4,1) is the only stationary point of f(x,y), no interior point (with strict inequalities 0 < x < 2 and 0 < y < 2) can possible be a maximum, local or otherwise. Therefore, as Mark44 has suggested, you need to look along the boundary lines x = ± 2 and/or y = ± 2 in order to locate a constrained maximum.
     
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