Minimizing sum of functions under constrained domain

Question

Consider this function, I am told to find minimum of it. I considered using lagrange multipliers but I have no constraint curves to work with.

Answer 1

An approach using coordinate geometry:

You can think about $2$ points, $P$ and $Q$ having coordinates $(x_1, x_1^2/20)$ and $\left(x_2,\sqrt{(17-x_2)(x_2-13)}\right)$ respectively.

You're required to find the square of the minimum distance between them.

$P$ lies on the parabola $y=x^2/20$

And after a bit of simplification, you can see that $Q$ lies on the circle $(x-15)^2+y^2 = 4$

Now our problem changes to figuring out the shortest distance between a circle and a parabola.

The line of shortest distance between these curves must be normal to both of them; so it must pass through the centre of the circle, $(15, 0)$

And if the line intersects the parabola at $(h, h^2/20)$, it's slope must be $-10/h$

Using these conditions, we get the equation of the line to be $$y = -\frac{10}{h}(x-15)$$ This passes through the point $(h, h^2/20)$ on the parabola, plugging this into the equation gives $h = 10$ as the only real solution.

The point on the parabola is then $A(10, 5)$ and that on the circle is $B(15-\sqrt2, \sqrt2)$

And the square of the distance between them is $2(5-\sqrt2)^2$