Constraining the parameter space of a function

[spaghetti3451]

Consider the function

$$f(x) = - \frac{1}{2}a^{2}x^{2} + \frac{1}{4}bx^{4} + d^{4}\cos(x/c),$$

where $a$, $b$, $c$ and $d$ are arbitrary parameters.

For some region(s) of the parameter space, there are oscillations in the function. My goal is to identify these regions of the parameter space.

Here are three examples:

Does this mean that there are bumps in the function only for $c \sim a$ (and any arbitrary value of $b$ and $d$)?

 

[mfb]

Did you calculate the derivative?
What does it tell you?

 

[spaghetti3451]

I find that

$$f'(x) = - ax + bx^{3} - \frac{d^{4}}{c}\sin(x/c).$$

Therefore, the minima are given by the solutions to the equation

$$ bcx^{3} - acx = d^{4}\sin(x/c).$$

Finding a closed-form solution to this equation is a big ask, I suppose. But the question is really about the number of solutions to this equation, and not the solutions themselves.

 

[mfb]

There is no closed-form solution, but you know that the right side is limited by $\pm d^4$.

 

[spaghetti3451]

There is no closed-form solution, but you know that the right side is limited by $\pm d^4$.

So, $bcx^{3} - acx = 0$ is a cubic equation with solutions at $x = 0, \pm \sqrt{\frac{a}{b}}$.

And, $d^{4}\sin(x/c)$ is a sinuosidal function with amplitude $d^{4}$ and period $2\pi c$.

How can I set up constraints now?

 

[mfb]

You can consider what happens for small d or large c, for example. In which region can it be relevant for the full equation?
How often does the sine term change its sign in this region?