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    Extreme Values, Differentials and Maximizing Areas

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    1) Find the absolute extreme values of the function
    f(x,y) = x^2 + xy - x - 2y + 4 on the region D enclosed by y= -x, x=3, y=0

    2) Given a circle of radius R. Of all the rectangulars inscribed in the circle, find the rectangular with the largest area.

    3) a) Find the differential df of f(x,y)= x(e^y)
    b) use the differential to estimate:
    (square root of 24)* (e^0.001)

    © BrainMass Inc. brainmass.com February 24, 2021, 2:14 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/extreme-values-differentials-maximizing-areas-9140

    Solution Preview

    1.)
    f(x,y) = x^2 + xy - x - 2y + 4
    partial derivative:
    del(f)/del(x) = 2x + y -1 = 0 (for extremum) ... (1)
    del(f)/del(y) = x - 2 = 0
    => x = 2 ..(2)
    From eqn 1 and 2: 2*2 + y - 1 = 0 => y = -3
    so, minima is at (2, -3) which is out of the enclosed region D.
    therefore check at the boundaries of given region (0,0), (3,0), (3,-3):
    (see figure)
    At x = 0, y =0
    => f(0,0) = 4
    at(3,0) :f(3,0) = 9 + 0 - 3 - 0 + 4 = 10
    at(3,-3): f(3,-3) = 9 + 9 - 3 - 6 + 4 = 13
    Therefore, the extreme of function are: 13 (maximum) at (3,-3) and 4 (minimum) at ...

    Solution Summary

    Extreme values, differentials and maximum area of a rectangle in a circle are found. The differentiation functions are analyzed.

    $2.19

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