# A Constraining the parameter space of a function

1. May 7, 2017

### 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 $d$)?

2. May 7, 2017

### Staff: Mentor

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

3. May 7, 2017

### 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.

4. May 7, 2017

### Staff: Mentor

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

5. May 7, 2017

### spaghetti3451

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?

6. May 8, 2017

### Staff: Mentor

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?