I want to find a solution to the following minimization problem using only elementary methods.
This is to say: high school algebra, basic inequalities, basic trigonometry and trigonometrical equations, Euclidean geometry, basic analytic geometry
but not any method that involves derivatives
problem: Consider the points $B(4,3)$, $C(6,1)$ and $M(2,k)$. Find the $k$ so the sum
$$ \lVert \vec{MB} \rVert\ + \lVert \vec{MC} \rVert $$
becomes the minimum.
my thoughts:
let's use a easier notation, so:
$$\lVert \vec{BC} \rVert\ = a $$
$$ \lVert \vec{MB} \rVert\ = b $$
$$\lVert \vec{MC} \rVert\ = c$$
and $$ \phi=\angle BMC $$
we want to find the minimum of $b+c$ or the minimum of $(b+c)^2$
but
$$(b+c)^2=b^2+c^2+2bc$$
The law of cosines for the triangle $BMC$ gives
$$a^2=b^2+c^2- 2bc \cdot \cos\phi$$
if we use both equations we want to find the minimum of
$$a^2+2bc(\cos\phi+1)$$
which implies that $$\cos\phi = -1$$
Is this method valid?
Do you have any other approach?
This can be thought of as a reflection problem. That is to say, reflect the point $A$ through the line $x=2$ (obtaining the point $\hat {A}=(0,3)$) and then draw the straight line segment connecting $\hat {A}$ to $C$. That segment crosses the line $x=2$ at the desired point $M$.