Question

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. $$ f(x, y)=x^{2 / 3}+y^{2 / 3} $$

Short answer

The function \( f(x, y)=x^{2 / 3}+y^{2 / 3} \) has only one critical point at the origin (0,0), for which the Second Partials Test fails.

Step-by-step solution

Compute the first partial derivatives

The first partial derivatives are: \( f_x = \frac{2}{3} x^{-1/3} \) and \( f_y = \frac{2}{3} y^{-1/3} \).

Locate the critical points

The critical points are where the first partial derivatives are zero or undefined. Here, \( f_x = 0 \) when \( x = 0 \), and \( f_y = 0 \) when \( y = 0 \). Hence, the only critical point of the function is at the origin (0,0).

Compute the second partial derivatives

We compute the second partial derivatives and get: \( f_{xx} = -\frac{2}{9} x^{-4/3} \), \( f_{yy} = -\frac{2}{9} y^{-4/3} \), and \( f_{xy} = 0 \).

Apply the Second Partials Test

With the Second Partials Test, we check \( D = f_{xx}f_{yy} - (f_{xy})^2 \) at the point (0,0). Here, \( f_{xx}(0,0) \) and \( f_{yy}(0,0) \) are undefined. Therefore, the Second Partials Test fails at the origin.