, 1. (Section 10.2)
Consider the R2 � R function defined by
f(x; y) = y
x
� ln x +
1
2y2:
Find the critical points of f and their nature.
[10]
𝑷𝒓𝒐𝒃𝒍𝒆𝒎 𝟏: 𝑪𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝑷𝒐𝒊𝒏𝒕𝒔 𝒐𝒇 𝒇(𝒙, 𝒚)𝒇(𝒙, 𝒚)
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏:
𝑓(𝑥, 𝑦) = 𝑦𝑥 − 𝑙𝑛 𝑥 + 12𝑦2𝑓(𝑥, 𝑦) = 𝑥𝑦 − 𝑙𝑛𝑥 + 21𝑦2
𝑺𝒕𝒆𝒑 𝟏: 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒑𝒂𝒓𝒕𝒊𝒂𝒍 𝒅𝒆𝒓𝒊𝒗𝒂𝒕𝒊𝒗𝒆𝒔.
𝜕𝑓𝜕𝑥 = −𝑦𝑥2 − 1𝑥𝜕𝑥𝜕𝑓 = −𝑥2𝑦 − 𝑥1𝜕𝑓𝜕𝑦 = 1𝑥 + 𝑦𝜕𝑦𝜕𝑓
= 𝑥1 + 𝑦
𝑺𝒕𝒆𝒑 𝟐: 𝑺𝒆𝒕 𝒕𝒉𝒆 𝒑𝒂𝒓𝒕𝒊𝒂𝒍 𝒅𝒆𝒓𝒊𝒗𝒂𝒕𝒊𝒗𝒆𝒔 𝒕𝒐 𝒛𝒆𝒓𝒐 𝒕𝒐
𝒇𝒊𝒏𝒅 𝒄𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝒑𝒐𝒊𝒏𝒕𝒔.
−𝑦𝑥2 − 1𝑥 = 0 ⇒ −𝑦 + 𝑥𝑥2 = 0 ⇒ 𝑦 = −𝑥 − 𝑥2𝑦 − 𝑥1 = 0
⇒ −𝑥2𝑦 + 𝑥 = 0 ⇒ 𝑦 = −𝑥1𝑥 + 𝑦 = 0 ⇒ 𝑦
= −1𝑥𝑥1 + 𝑦 = 0 ⇒ 𝑦 = −𝑥1
𝑺𝒕𝒆𝒑 𝟑: 𝑺𝒐𝒍𝒗𝒆 𝒕𝒉𝒆 𝒔𝒚𝒔𝒕𝒆𝒎 𝒐𝒇 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔.
Consider the R2 � R function defined by
f(x; y) = y
x
� ln x +
1
2y2:
Find the critical points of f and their nature.
[10]
𝑷𝒓𝒐𝒃𝒍𝒆𝒎 𝟏: 𝑪𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝑷𝒐𝒊𝒏𝒕𝒔 𝒐𝒇 𝒇(𝒙, 𝒚)𝒇(𝒙, 𝒚)
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏:
𝑓(𝑥, 𝑦) = 𝑦𝑥 − 𝑙𝑛 𝑥 + 12𝑦2𝑓(𝑥, 𝑦) = 𝑥𝑦 − 𝑙𝑛𝑥 + 21𝑦2
𝑺𝒕𝒆𝒑 𝟏: 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒑𝒂𝒓𝒕𝒊𝒂𝒍 𝒅𝒆𝒓𝒊𝒗𝒂𝒕𝒊𝒗𝒆𝒔.
𝜕𝑓𝜕𝑥 = −𝑦𝑥2 − 1𝑥𝜕𝑥𝜕𝑓 = −𝑥2𝑦 − 𝑥1𝜕𝑓𝜕𝑦 = 1𝑥 + 𝑦𝜕𝑦𝜕𝑓
= 𝑥1 + 𝑦
𝑺𝒕𝒆𝒑 𝟐: 𝑺𝒆𝒕 𝒕𝒉𝒆 𝒑𝒂𝒓𝒕𝒊𝒂𝒍 𝒅𝒆𝒓𝒊𝒗𝒂𝒕𝒊𝒗𝒆𝒔 𝒕𝒐 𝒛𝒆𝒓𝒐 𝒕𝒐
𝒇𝒊𝒏𝒅 𝒄𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝒑𝒐𝒊𝒏𝒕𝒔.
−𝑦𝑥2 − 1𝑥 = 0 ⇒ −𝑦 + 𝑥𝑥2 = 0 ⇒ 𝑦 = −𝑥 − 𝑥2𝑦 − 𝑥1 = 0
⇒ −𝑥2𝑦 + 𝑥 = 0 ⇒ 𝑦 = −𝑥1𝑥 + 𝑦 = 0 ⇒ 𝑦
= −1𝑥𝑥1 + 𝑦 = 0 ⇒ 𝑦 = −𝑥1
𝑺𝒕𝒆𝒑 𝟑: 𝑺𝒐𝒍𝒗𝒆 𝒕𝒉𝒆 𝒔𝒚𝒔𝒕𝒆𝒎 𝒐𝒇 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔.
