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Extrema of Functions of 2 Variables- HW Problems
Find the critical points for the following functions.
1. 𝑔(𝑥, 𝑦) = −𝑥 2 + 𝑦 2 + 𝑥𝑦
2. 𝑓(𝑥, 𝑦) = 2𝑥 3 − 6𝑥𝑦 + 𝑦 2
3. ℎ(𝑥, 𝑦) = 𝑥 2 𝑦 − 𝑥𝑦 2 − 𝑥 + 𝑦
4. 𝑓(𝑥, 𝑦) = −𝑥 3 − 𝑦 2 + 2𝑥𝑦 + 3
Find all local maxima, minima, and saddle points for the following
functions.
5. 𝑓(𝑥, 𝑦) = 2𝑥 3 − 6𝑥𝑦 + 𝑦 2
6. 𝑔(𝑥, 𝑦) = 2𝑦 3 − 6𝑥𝑦 + 3𝑥 2
7. Find the points on the surface 𝑦 2 = 𝑥𝑧 + 9 that are closest to the
origin.
8. A rectangular box has a volume of 64𝑚3 . Find the dimensions of
the box with minimum surface area.
Extrema of Functions of 2 Variables- HW Problems
Find the critical points for the following functions.
1. 𝑔(𝑥, 𝑦) = −𝑥 2 + 𝑦 2 + 𝑥𝑦
2. 𝑓(𝑥, 𝑦) = 2𝑥 3 − 6𝑥𝑦 + 𝑦 2
3. ℎ(𝑥, 𝑦) = 𝑥 2 𝑦 − 𝑥𝑦 2 − 𝑥 + 𝑦
4. 𝑓(𝑥, 𝑦) = −𝑥 3 − 𝑦 2 + 2𝑥𝑦 + 3
Find all local maxima, minima, and saddle points for the following
functions.
5. 𝑓(𝑥, 𝑦) = 2𝑥 3 − 6𝑥𝑦 + 𝑦 2
6. 𝑔(𝑥, 𝑦) = 2𝑦 3 − 6𝑥𝑦 + 3𝑥 2
7. Find the points on the surface 𝑦 2 = 𝑥𝑧 + 9 that are closest to the
origin.
8. A rectangular box has a volume of 64𝑚3 . Find the dimensions of
the box with minimum surface area.
