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Extreme Values, Differentials and Maximizing Areas

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)

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.

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