A huge love for 1729.

The $LHS$ is a sum of two exponents ${ 17 }^{ \sqrt { x-1729 } }$ and ${ 29 }^{ \sqrt { x-1729 } }$ and is increasing since the bases of the exponents are greater than $1$ .

Combining this with the fact that $\min { \left\{ { 17 }^{ \sqrt { x-1729 } }+{ 29 }^{ \sqrt { x-1729 } } \right\} } =2>0.1729$ at $x=1729$ we can see that there are no points of intersection of those functions.

Square root will yield a positive quantity. Therefore the powers of both will be positive and hence both of the quantities will be greater than once and hence there minimum value will be 2 at x = 1729