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I have an inequality as below, and I need to find the solution set of this inequality. I couldn't find a method on how to solve it. It would be great if I could find all the $L_1$ and $L_2$ values that satisfy the inequality. There are conditions for $L_1$ and $L_2$ values in optimization methods. Thus, I cannot use these methods. Even if there is a suitable method, I couldn't find it. Additionally, I tried to find a solution based on the function's graph inside the absolute value, but there is no site/program that can plot equations with arbitrary constants $L_1$ and $L_2$, etc., or I couldn't find it. Can you help me with these issues?

$$\sup_{x\in[0,1)\cup(1,2]}\{|x^2-1-(L_1(x-1)+L_2(x-1)^2)|\}\le 2$$

Note: The number on the right side of the inequality can be changed. Instead of 2, a positive real number can be written.

Burak
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  • Hint (If this has been discussed in your class), try to use the Cauchy-Schwarz inequality. – Zim Jun 14 '23 at 08:55
  • Also, please include your attempts at solving the problem in your post. – Zim Jun 14 '23 at 08:55
  • I know the Cauchy-Schwarz inequality, but this inequality is not enough to find all $L_1$ and $L_2$ values. I am trying to find all $L_1$ and $L_2$ values. This problem is a problem I encountered during my doctoral studies. – Burak Jun 14 '23 at 09:02
  • Just some thoughts: If the absolute value were not present, then the dominant coefficient of $x^2$ is $(1-L_2)$. This means that the polynomial is convex if $1>L_2$ and concave if $1<L_2$. If convex, then its extrema would occur at the endpoints, so it would suffice to check the value at $x=0$ and $x=2$. If it is concave, it would suffice to check at the critical point (in $x$) since that will be the function's global maximum. However, the absolute-value throws a bit of a wrench in this process and will probably require more case analysis. – Zim Jun 15 '23 at 07:41
  • Thank you, @Zim. I understand your thoughts. Finally, do you know of a program/site that can automatically plot the absolute value inequality according to $L_1$ and $L_2$? So, the program/site should give the graphs according to different $L_1$ and $L_2$ values; I shouldn't provide those values. Does such a program/site exist? – Burak Jun 15 '23 at 08:10
  • desmos might help, but in general no, you'd have to derive it yourself. – Zim Jun 15 '23 at 08:49
  • Thank you so much. – Burak Jun 15 '23 at 13:06

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