Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

2 Answers 2

up vote 3 down vote accepted

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

share|improve this answer
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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.