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    Finding maxima and minima of a mathematical function

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    Find the local maximum and minimum values and the saddle points of f(x,y) = 2xy + x-y.

    © BrainMass Inc. brainmass.com December 24, 2021, 4:57 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/finding-maxima-and-minima-of-a-mathematical-function-18497

    SOLUTION This solution is FREE courtesy of BrainMass!

    A function of two variables has a local maximum at (a,b)
    <br>if f(x,y) =&lt; f(a,b) when (x,y) is near (a,b)
    <br>
    <br>The number f(a,b) is called a local maximum value.
    <br>
    <br>If f(x,y) &gt;= f(a,b) when (x,y) is near (a,b) then f(a,b) is a local minimum value.
    <br>
    <br>
    <br>our function is,
    <br>
    <br>f(x,y) = 2xy + x -y
    <br>
    <br>let me use fx = partial derivative with respect to x and fy for y
    <br>
    <br>fx = 2 y + 1 -----(1)
    <br>
    <br>fy = 2x - 1 -----(2)
    <br>
    <br>to find the crical point, solve the equations,
    <br>
    <br>2y+ 1 = 0 and 2x-1 = 0
    <br>
    <br>2y+1=0 ====&gt; y = -1/2
    <br>
    <br>2x-1=0 ====&gt; x = 1/2
    <br>
    <br>Therefore, the critical point is (1/2, -1/2)
    <br>
    <br>Now to the second derivative test
    <br>
    <br>D = D(a,b) = fxx (a,b) * fyy(a,b)- [fxy (a,b)]^2
    <br>(Please see the attached figure)
    <br>
    <br>fxx(a,b) = 0 (differentiate eq 1 with respect to x again)
    <br>fyy(a,b) = 0
    <br>
    <br>fxy = 2 (differentiate eq 1 with respect to y)
    <br>
    <br>at(1/2, -1/2)
    <br>
    <br>D = 0 - 2 = -2
    <br>
    <br>Now refer to the figure, (1/2, -1/2) is a saddle point

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 4:57 pm ad1c9bdddf>
    https://brainmass.com/math/calculus-and-analysis/finding-maxima-and-minima-of-a-mathematical-function-18497

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