Locate and classify all critical points for the given function.
-$f(x, y)=x^{3}-5 x y+y^{3}$
Only find the critical points for this function.
Do not test for maximum or minimum.
A) $(0,0)$
B) $\left(\frac{5}{3}, \frac{5}{3}\right)$
C) $(0,0)$ and $\left(\frac{5}{3}, \frac{5}{3}\right)$
D) There are no critical points.

Question

Quizplus · Accepted Answer

Critical points occur where the gradient is zero. Solving $\frac{\partial f}{\partial x} = 3x^2 - 5y = 0$ and $\frac{\partial f}{\partial y} = -5x + 3y^2 = 0$ gives the points $(0,0)$ and $\left(\frac{5}{3}, \frac{5}{3}\right)$.

Locate and Classify All Critical Points for the Given Function f(x,y)=x3−5xy+y3f(x, y)=x^{3}-5 x y+y^{3}f(x,y)=x3−5xy+y3

Locate and Classify All Critical Points for the Given Function $f(x, y)=x^{3}-5 x y+y^{3}$