Minizing the total lenght of 2 pipelines, Optimization question

Question

A pump house is to be placed at some point $X$ along a river, A pipe from point $A$ and a pipe from point $B$ will then be connected to the point $X$. How far should $X$ be away from $M$, so that the total length of the pipes $\overline{AX}$ and $\overline{BX}$ are minimised?

I just need help with setting up the function.

Answer 1

$AM+XB$ is minimum when reflexing $B$ in $D$ wrt $MB$ we connect $AD$. The point $X$ where $AD$ intersects $MN$ is the minimum required. Because any other point $P$ is such that $AP+PD>AD$ for the triangular inequality.

$\triangle ACD$ similar $\triangle AMX$

$MX:AM=CD:AC$

$MX=\frac{AM\cdot CD}{AC}=\frac{10}{3}$km

Answer 2

Here's what I did.

Hyputenuse of triangle AMX= sqrtroot of 4+×^2

Hyputenuse of triangle BNX= sqrtroot of 1+(5-x)^2

Sqrt root 4+x^2 + sqrtroot 1+(5-x)

I differentiate that and got (×/sqrroot 4+x^2) - (-x+5/sqrtroot x^2 - 10x +26)

I graphed the function and got 3.333 I then made a sign diagram which clarified that It was a local minimum