Question

given a function of two variables, $z = f(x, y)$, and a point $p_0: (x_0, y_0)$: if $p_0$ is a local maxima or minima of $f$, and not on the boundary of the domain of $f$, then $p_0: (x_0, y_0)$ is a critical number of $f$. \bigcirc true \bigcirc false

Explanation:

For a function \( z = f(x,y) \) of two variables, a critical point (or critical number here) is a point \( (x_0,y_0) \) where either both partial derivatives \( f_x(x_0,y_0) = 0 \) and \( f_y(x_0,y_0)=0 \), or at least one of the partial derivatives does not exist. If a point \( P_0=(x_0,y_0) \) is a local maximum or minimum (and not on the boundary of the domain), then by the theorem for local extrema of functions of two variables, the partial derivatives at \( P_0 \) must be zero (assuming the function is differentiable there, which is a common case in such theorems). So if \( P_0 \) is a local max/min and not on the boundary, it must satisfy the conditions for a critical point (either partial derivatives are zero or don't exist; for local extrema in the interior of the domain, differentiable functions will have partial derivatives zero at such points). Thus the statement is True.

Answer:

True