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Consider the function $$f(x,y)=(x+y)^4$$ and determine whether $f$ has a maximum, a minimum or neither at the point $(0,0)$.

I thought I needed to use the second partial derivative test but how would I go about showing the point is neither a minimum nor a maximum if the test is inconclusive?

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up vote 3 down vote accepted

Hint: what can you say about the fourth power of any real number?

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It's always positive? I'm just stuck on how to go about proving that the point is neither a minimum nor a maximum. – user51462 Dec 13 '12 at 7:43
The functions plot shows z=0 where y=-x – user51462 Dec 13 '12 at 7:50
It's always $\ge 0$. Surely you don't need a plot to see that $(x+y)^4 = 0$ when $y=-x$. So if the function is always $\ge 0$, and it's equal to $0$ at $(0,0)$, is $(0,0)$ a minimum, a maximum, or neither? – Robert Israel Dec 13 '12 at 7:52
Sorry, the question asked me to refer to the plot as well. I thought it was neither because z=0 along the line y=-x so it can't be an extrema? – user51462 Dec 13 '12 at 8:02
What's the definition of an extremum? – Robert Israel Dec 13 '12 at 8:52

$$f(0, 0) = 0$$

What other values can $f$ assume? For example, at $(1,1)$, $(-1, 1)$, etc.

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