Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

226_Functions of Several Variables_[Ch. 6
Test for maximum and minimum the following functions of
two variables:
2008. z = (x \)*+2y*.
2
2009. z = (x I)
2y*.
2010. z = x* + xy + tf 2x y.
2011. z = *y (6 xy)(x>0 9
2012. 2 = x 4 + y 4
2013. z = x
2014. z=l
2015. z = (
20,6. 2 =
2x*+4xy 2y z
.
Find the extrema of the following functions of three variables:
2017. a = x 2 2
+ f/ +z 2
xy + x 2z.
2018. =^++ + -(^>0, y>0, z>0).
tions:
Find the extrema of the following implicitly represented func-
2019*. x* + y* + z* 2x+4y 62 11=0.
2020. x 9
y*3x + 4y + z* + z 8 = 0.
Determine the conditional extrema of the following functions:
2021. z = xy for *
2022. z^=x + 2y for
2023. z = x* + if for + -=
2024. z = cos 2
A: + cos a
f/ for// A: = ~-.
2
2025. W = A: 2y + 2z for A:
2026. ^ = ^ 2 2
+ f/ + 2 8
for
+ r
- + -
2027. u = xtfz* for .v-f-y + z= 12(jc>0,
2028. u^xyz provided x-(~j/+e=5, xy+yz+zx=8.
2029. Prove the inequality
if x^zQ
Hint: Seek the maximum of the function u = xyz provided