Subtopic: Maxima and minima of function with several independent variables
Question Statement:
For the multivariable function $f(x, y) = x^3 + y^3 - 3xy$, the non-zero critical point is $(a, b)$. The local minimum value attained by $f(x, y)$ at this point is ________ (answer in integer).
Answer Key:
-1
Step-by-Step Solution & Explanation
**Step 1: Determine stationary points**
$\frac{\partial f}{\partial x} = 3x^2 - 3y = 0 \implies y = x^2$
$\frac{\partial f}{\partial y} = 3y^2 - 3x = 0 \implies x = y^2$
Substituting $y = x^2$ into the second equation: $x = (x^2)^2 = x^4 \implies x(x^3 - 1) = 0$.
Real roots are $x = 0$ (giving $(0,0)$) and $x = 1$ (giving non-zero critical point $(1,1)$).
**Step 2: Second derivative test at $(1, 1)$**
$r = f_{xx} = 6x = 6 > 0$
$s = f_{xy} = -3$
$t = f_{yy} = 6y = 6$
$D = rt - s^2 = (6)(6) - (-3)^2 = 36 - 9 = 27 > 0$
Since $D > 0$ and $r > 0$, $(1, 1)$ is a point of local minimum.
**Step 3: Minimum value**
$f(1, 1) = 1^3 + 1^3 - 3(1)(1) = 1 + 1 - 3 = -1$