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...

Answer 1

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:

1. 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.
2. 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.