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Finding maximum value of $\bigg|\sqrt{x^2-8x+52}-\sqrt{x^2-4x+8}\,\bigg|$

What I try is to write the function as

$\bigg|\sqrt{(x-4)^2+(0-6)^2}-\sqrt{(x-2)^2+(0-2)^2}\,\bigg|$

Now we have to maximize difference of distance between point $P(x,0)$ and $A(4,6)$ and $(2,2)$.

i.e. we have to maximize $|PA-PB|$

How do I solve it ? Please, have a look.

Angelo
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jacky
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  • I tried differentiating $f(x) = \displaystyle \bigg|\sqrt{x^2-8x+52}-\sqrt{x^2-4x+8}\bigg|$, setting it equal to $0$ to find the value of $x$ which gives the maximum/minimum, then plugging in those values of $x$ back into $f.$ I think it works... – Adam Rubinson Aug 14 '23 at 09:19

1 Answers1

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It is not necessary to differentiate any function, indeed you can use the reverse triangle inequality.

If $P(x,0)$ does not belong to the line $r$ passing through $A(2,2)$ and $B(4,6)$, we get the triangle $ABP$, consequently, by applying the reverse triangle inequality, it results that

$|PA-PB|<AB.$

Whereas, if $P(x,0)$ belongs to the line $r$, then $|PA-PB|=AB.$

Hence the maximum is $AB=2\sqrt5\,.$

Angelo
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