Question

Finding the critical points, what happens if (df/dx)²+(df/dy)² never equals to 0?

a) The function has no critical points
b) The function has infinitely many critical points
c) The function has exactly one critical point
d) The critical points cannot be determined without additional information

Answer 1

If (df/dx)²+(df/dy)² is never zero, the function has no critical points because the gradient never equals zero at any point, implying the absence of points where the derivative is zero.

Step-by-step explanation:

If the sum of the squares of a function's partial derivatives, (df/dx)²+(df/dy)², never equals to 0, it implies that neither partial derivative can simultaneously be zero. This means that for the function in question, the gradient is never zero, and thus, there are no points where the first derivatives in both the x and y directions are zero at the same time. Therefore, the function has no critical points. Critical points are locations where the first derivative(s) of a function are zero or not defined, which are candidates for local maxima, local minima, or saddle points.