Determine the global extreme values of the function on the given set without using calculus.
Recall the existence of global extrema. Let f(x, y) be a continuous function on a closed, bounded domain . Then f(x, y) has both a minimum and a maximum value on .
The given domain is D = {(x, y) | x2 + y2 ≤ $a}.
This domain is lNebL7Gfb9CLXY77oThJxuuCacaeJ5ts/AEB96iXO4Ct9MOJ7+hqC6E5PhJPLcSxctvxH5NThydjow5DfD364FKP0fhp3bXLNHUFGA==. Also, f(x, y), being an exponential function, bfdtn5PFpMarVC8x2zfiEQ== continuous.
Thus f(x, y) zVpudauK/UPx/KjQc3EOeIN8nEc= attain both a maximum and minimum on D.
Write
Observe that f(x, y) will attain its maximum value when attains its CoKdRWMwar7l2TXZp/HL1V0+ugY= value. This will occur when x2 + y2 attains its CoKdRWMwar7l2TXZp/HL1V0+ugY= value. Since x2 ≥ 0 and y2 ≥ 0, this value of x2 + y2 is x2 + y2 = 1Wh3cvJ2xF4=.
Find the maximum value of f(x, y).
0VV1JcqyBrI=
Write
Observe that f(x, y) will attain its minimum value when attains its 3zePInBsBgbXNBfSatFas/mjlxc= value. This will occur when x2 + y2 attains its 3zePInBsBgbXNBfSatFas/mjlxc= value. Since x2 ≥ 0 and y2 ≥ 0, this value of x2 + y2 is x2 + y2 = nc1ItEz0kR4=.
Find the minimum value of f(x, y).
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