Algebraic Manipulation !

Let's Do some Geometric , An Smart Geometrical Interpretation Say That This f(x) is written as Sides of an Triangle in terms of Cosine Law , Such That It forms an right angle triangle ABC ( See Figure ) :

So Geometrically : $f\left( x \right) =AD+CD$ . By using Triangle Inequality ;

$\because \quad AD+CD\quad \ge \quad AC\\ \therefore \quad { f\left( x \right) }_{ min }\quad =\quad AC$ .

Using Pythagoras Theorem in Triangle ABC we get ' $AC=25$ ' So Using ''Equality'' Condition of ''Triangle inequality'' that Sum of Two side is equal to third Side Then All Three Point's must be Collinear , So Redraw It's figure :

Now Use : $Area(\Delta ADB)+Area(\Delta CDB)=Area(\Delta ABC)\\ \\ \cfrac { 1 }{ 2 } (7x)\sin { 30 } +\cfrac { 1 }{ 2 } (24x)\sin { 60 } =\cfrac { 1 }{ 2 } (7)(24)\sin { 90 } \\ \\ \cfrac { 336\quad -\quad 24\sqrt { 3 } x\quad -\quad 7x }{ 4 } \quad =\quad 0\\ \\ \boxed { E=0 }$ .

Q.E.D