We tend to be taught to interpret the gradient to be the direction of steepest ascent of a vector valued function.
However, I'm confused as to why when we reach the minimum of a function, say f(x,y) = x^2 + y^2, we no longer expect the gradient to tell us how to increase. We're at a minimum,so certainly this instrument that tells us how which direction we should move in so that we increase the value of the function should be useful, no? Instead it fails to tell us where to go to increase and just yields a zero vector.
Should I interpret this as meaning that the interpretation of the gradient as telling us the direction of steepest ascent breaks down at the minimum/maximum? That this is simply not always true and that its truest definition is that it represents rates of change? Does this have a deeper mathematical meaning?