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I have an optimization problem:

$$f(x)=\frac{\sqrt{\pi}}{\varphi \cosh (x \pi \sqrt{2}) \exp \left(x \mathbb{1}_{\{x>0.5\}}\right)+0.625}+0.05\left|\sin \left(\pi^3(x+0.1)^{1.1}\right) \exp (-\pi|0.8-x|)\right|$$

where $\varphi = \frac{1+\sqrt{5}}{2}$.

My problem is to optimize $f(x)\cdot x$, my thought was to use the derivative: $f'(x)\cdot x + f(x)$, however, I noticed the indicator in the exponential-function, making the function probably not continuous, and then not differentiable. (The use of absolute value is problematic for differentiability)

I have tried to insert the function in Desmos, and found this

enter image description here

However, I need to find the result without using any CAS-software. Is there any hints or somewhat that can get me in the right direction?

Rodrigo de Azevedo
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Overkill123
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1 Answers1

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Let me write $g(x) := x f(x)$. Here are some hints.

  • Concerning the lack of regularity. You have different sources of singularities. First, as you noted, the indicator $\mathbf{1}_{\{x>0.5\}}$ causes a discontinuity at $x = 0.5$. Second, you have an absolute value $|x-0.8|$, which, while it is continuous, is not differentiable at $x = 0.8$. Third, you also have an absolute value of a sine, which is thus not differentiable at $\pi^3 (x+0.1)^{1.1} = k \pi$ for $k \in \mathbb{Z}$. Of course, these are difficulties. But, within each interval, you can apply the criteria $g' = 0$ to look for a local maximum. This will give you "candidates" to be the global maximum.

  • Concerning the range of $x$. Your function $f$ is the sum of two terms, both of which decay like $e^{-x}$ as $x \to +\infty$. This, the function $g(x) = x f(x)$ decays like $x e^{-x}$ so is small for large $x$. Such bounds can help you restrict the list of candidates.

cs89
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