Is there any way we can test if a function has local optima or not without searching?

Question

Suppose, we have the following function $f(x)$:

Is there any way, without searching, we can test if this function has local optima or not?

Can we, say, test any property of this function, or, do some calculation to know this beforehand?

Note: Searching means applying some searching algorithms like the Bisection method, Newton's method, and so on.

Answer 1

It is a very well-known and often used fact that a differentiable function $g(x)$ satisfies

$g'(y) = \dfrac{dg(y)}{dy} = 0 \tag 1$

at any local maximum or minimum $y$; see this wikipedia page. If we apply this principle to the function at hand, which is a fourth-degree polynomial

$f(x) = ax^4 + bx^3 + cx^2 + dx + e \in \Bbb R[x], \tag 2$

we see that the extrema occur at those $y \in \Bbb R$ such that

$4ay^3 + 3by^2 + 2cy + d = f'(y) = 0; \tag 3$

since $f'(x)$ is a cubic, there is available to us an explicit method for finding its roots; therefore we can find the extrema of any quartic, real polynomial. Similarly, given a quintic polynomial $q(x)$, $q'(x)$ is a quartic to which we can apply algebraic methods; but if

$\deg f(x) \ge 6 \tag 4$

then

$\deg f'(x) \ge 5, \tag 5$

and there is in general no "algebraic method" for finding the zeroes of $f'(x)$. In the absence of such procedures, we are forced to turn to iterative techniques such as Newton's method, regula falsi, bisection, and so forth to find the zeroes of $f'(x)$. And having found them, we can of course evaluate $f''(x)$ to try and discover if they are maxima, minima, or inflection points.

Answer 2

I am assuming you have some basic analysis knowledge: To find potential critical points you determine the derivative of your, in this case polynomial, function f and then you compute the zeroes $x_i$ of $f'$. These zeroes are potential (local) minimizers\maximizers. If $f''(x_i) > 0$ then the function has a min. in this point.

Answer 3

If we're testing for whether the function has local extrema, we don't necessarily have to go all the way to finding zeros of the derivative. You see, extrema happen where the derivative changes sign - if we have a positive derivative somewhere, and a negative derivative somewhere else, there's a maximum or minimum (depending on the order of those two points) between them. This so-called "first derivative test" is completely reliable.

So then, based on this, we don't have to keep iterating very long. A shallow search may be necessary, but we know there's an extremum as soon as we find the derivative changing sign.