I don't understand your reasoning. You calculated the second derivatives along two different paths and got -2 and 2. What makes you sure that those are the smallest and largest values?
What if we approach (0, 0) along the line y= 10000x? Then . What is the second derivative of that?
Do it more generally. Since the derivative is "local", we can approximate any path through (0, 0) by a straight line: y= mx for some number, m. Then . What is the second derivative of that?
That includes every path except vertical ones- on a line tangent to such a path, at (0, 0), x= 0 so f(x,y)= f(0, y)= 0 and the second derivative is 0.