Here is the problem from recently concluded AIME 2021.
Find the number of integers c such that the equation $||20|x|-x^2|-c|=21$ has 12 distinct real solutions.
I know how one can solve these algebraically. However I have always found (my personal preference) that plotting the graph of the following equation: $||20|x|-x^2|= c+21$ and $||20|x|-x^2|= c-21$ and then getting a pictorial understanding a much more elegant technique.
However under the exam situation I find it difficult to plot the above 2 graphs. My question to valuable members to guide me on the following points:
- How do I quickly get critical points to sketch the graph? Is there any good resource which will help me in mastering the graphing of various special functions?
- Can the algebraic way be simplified for newbies like me to ensure clear understanding?
You can go on unpacking the absolute value signs. When you do, you should note what assumption implies each sign. The innermost one is easy. If $x \ge 0$ you have $|20x-x^2|$, which is $20x-x^2$ for $0 \le x \le 20$ and $x^2-20x$ for $x \gt 20$. Similarly if $x \lt 0$ you have $|-20x-x^2|$ which is $-20x-x^2$ for $-20 \le x \lt 0$ and $x^2+20x$ for $x \lt -20$