Find a point $P$ on the graph of $$x^2+y^2- 142 x - 10 y + 4841 = 0$$ and a point $Q$ on the graph of $$(y - 5)^2 = x^3 - 155 x^2 - 274 x + 583248$$ such that the distance between them is as small as possible.

To solve this problem, we let $(x,y)$ be the coordinates of the point $Q$. Then we need to minimize the following function of $x$ and $y$:

After we eliminate $y$ from the above, we reduce to minimizing the following function of $x$ alone:
$f(x)=$
To find the minimum value of $f(x)$ we need to check the value at the following three points (in increasing order). (You will need to use a numerical method, like Newton-Raphson to find one of these points.)
$x_1=$
$x_2=$
$x_3=$
We conclude that the minimum value of $f(x)$ occurs at
$x=$
Thus a solution to our original question is
$P= ($$,$$)$
$Q= ($$,$$)$

Hint: