This statement is False.

For a function $f(x, y)$, the conditions $f_x(a, b) = 0$ and $f_y(a, b) = 0$ (i.e., both first partial derivatives are zero) do indicate that $(a, b)$ is a critical point. However, this point could be:

1. A local minimum
2. A local maximum
3. A saddle point (neither a minimum nor a maximum)

To determine whether the point is a local minimum, maximum, or saddle point, one must check the second partial derivatives using the Hessian determinant:

$D = f_{xx}(a, b)f_{yy}(a, b) - \left( f_{xy}(a, b) \right)^2$

• If $D > 0$ and $f_{xx}(a, b) > 0$, the point is a local minimum.
• If $D > 0$ and $f_{xx}(a, b) < 0$, the point is a local maximum.
• If $D < 0$, the point is a saddle point.
• If $D = 0$, the test is inconclusive.

Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:

1. What is the role of the second derivative test in finding local extrema in multivariable functions?
2. How do saddle points differ from local minima and maxima?
3. Can a critical point where both partial derivatives are zero ever be a global maximum or minimum?
4. How do we compute the Hessian matrix for a given function $f(x, y)$?
5. What happens if the Hessian determinant $D = 0$? How do we interpret such a case?

Tip: Always verify the nature of a critical point using the second derivative test to avoid misclassifying saddle points!