Finding all the minima in a unconstrained minimization problem

Question

I've just read a textbook stating that the first order condition $\frac{df}{dx} = 0$ and second order sufficient condition $\frac{d^{2}f}{dx^2} \gt 0$ of unconstrained minimization will find the all the local minima.

Is this statement right?

Using only these conditions will miss the minimum point in case such as $f(x)=x^4$.

I'm confused...

Certainly $f^\prime = 0$ and $f^{\prime\prime} > 0$ fail to discover the point $x=0$ for $f(x) = x^4$. Does your book define a local minimum in some peculiar way such that "flat" minima like this aren't "local minima"? — Jason Zimba, Mar 23 '14 at 23:20
No. It does not say about that kind of restriction. Thank you. — user89855, Mar 23 '14 at 23:25

score 0 · Accepted Answer · answered Mar 23 '14 at 23:25

0

Yes, this is a case where the test is inconclusive (see here). In that case (according to the same source) you can use further criteria such as the Taylor expansion to determine the behaviour of $f(x)$ near any critical point. In, the case of the $f(x)$ that you mention:

1st order condition $$\frac{\partial f(x)}{\partial x}=4x^3$$ and this is equal to zero we find that $x=0$. Thus $x=0$ is a critical point.
2nd order condition $$\frac{\partial^2 f(x)}{\partial x^2}|_{x=0}\large=12x^2|_{x=0}=12\cdot0=0$$ So in this case the test is inconclusive.

But, since $f(x)=x^4$ is always nonnegative and for $x=0$ you have that $f(0)=0$ then you obtain that way the minimum.

answered Mar 23 '14 at 23:25

Jimmy R.

35,868

So... the statement is wrong. isn't it? – user89855 Mar 23 '14 at 23:30
@user89855 No it is not. Notice that is says sufficient condition and not necessary. Sufficient means that when it holds, then it is ok, but necessary means that if does not hold then it is not ok – Jimmy R. Mar 23 '14 at 23:30
The first sentence in the question. – user89855 Mar 23 '14 at 23:32
@user89855 Yes, ok. Is the explanation ok, see the previous comment, I edited it. – Jimmy R. Mar 23 '14 at 23:32
So... Using first order condition and second order sufficient condition CANNOT find ALL minima, right? – user89855 Mar 23 '14 at 23:39
@user89855 Right. – Jimmy R. Mar 23 '14 at 23:40
@user89855 you are welcome mate! – Jimmy R. Mar 23 '14 at 23:43

Finding all the minima in a unconstrained minimization problem

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