Are you using Mathematica or Wolfram|Alpha? If you are using Mathematica then compute the problem using the function Minimize as in

Minimize[Sqrt[c^2 + (b - x)^2] + Sqrt[a^2 + x^2], x]

if you are using Wolfram|Alpha then the query would be

minimize sqrt { { a }^{ 2 }+{ x }^{ 2 } } + sqrt { { (b-x) }^{ 2 }+{ c }^{ 2 } } with respect to x

however, it turns out that Wolfram|Alpha does not parse this properly. If one instead only uses parentheses for the expression as in

minimize sqrt ( ( a )^( 2 )+( x )^( 2 ) ) + sqrt ( ( (b-x) )^( 2 )+( c )^( 2 ) ) with respect to x

then Wolfram|Alpha parses it correctly but says that there are no global minima and does not return a result. However there are global minima for this function for various values of a,b, and c.