# 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)

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.