Relative Minimums and Maximums — Question 9

Question 9 – Solution

Strategy. Use the definition of a relative minimum first, then test each partial derivative with a signed two-sided limit.

Step 1: Direct extremum test At the origin, $f(0,0)=0$. For every $(x,y)\ne(0,0)$, $$f(x,y)=\sqrt{x^2+y^2}>0=f(0,0).$$ Therefore $$\boxed{(0,0)\text{ is a strict relative minimum}.}$$

Step 2: Partial derivatives Along the $x$-axis, $$\frac{f(h,0)-f(0,0)}h=\frac{|h|}{h},$$ whose right-hand limit is $1$ and left-hand limit is $-1$. Thus $f_x(0,0)$ does not exist. The identical calculation on the $y$-axis shows that $f_y(0,0)$ does not exist.

Step 3: Methodological conclusion Fermat’s theorem says that a differentiable interior extremum has zero gradient. It does not say every extremum is differentiable. Consequently, a complete candidate search must include interior points where first partial derivatives vanish and points where they fail to exist. This example supplies the latter case.