Find all local maximum and minimum points of the function $f$.

Question

Any help with this problem I have would be so much appreciated, i've been going round in circles and have no idea what I'm really doing.

I have the problem:

• Find all local maximum and minimum points of the function $f = xy$.

I've done the first derivative to get $$f' = 1(\frac{dy}{dx})$$

But I have no clue on how to find the local max and min from this. Any help would be grateful.

Thank you.

Answer 1

The local extrema of a function occur at stationary points, i.e. where the function's total derivative is $0$. In terms of partial derivatives, if the function has continuous partial derivatives (which this one does), this is equivalent to the partial derivatives both being $0$ at the point.

Partially differentiating, we have \begin{align*} f_x &= y \\ f_y &= x. \end{align*} These are both $0$ only at $(x, y) = (0, 0)$.

However, this stationary point is neither a local minimum nor maximum. Consider, for $t \in \Bbb{R} \setminus \{0\}$, \begin{align*} f(t, t) &= t^2 > 0 = f(0, 0) \\ f(t, -t) &= -t^2 < 0 = f(0, 0). \end{align*} That is, there are points as close as you want to $(0, 0)$ that have greater and lesser function values than $f(0, 0) = 0$.