<s>[INST] If $x$ is a nonnegative real number, find the minimum value of $\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$ If $x$ is a nonnegative real number, then find the minimum value of
$$\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$$
How can I approach this? Thanks
 [/INST] Process 1: Derivate it to zero
Process 2: Using triangle inequality,
Given,$$\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$$
$$\rightarrow\sqrt{x^2+2^2}+\sqrt{\left(-x+12\right)^2+3^2}\geq \sqrt{\left(x-x+12\right)^2+(2+3)^2} = \sqrt{{12}^2+{5}^2}$$
$$=\sqrt{169}=13$$
I.e, So,
The minimum value is $$\bbox[5px,border:2px solid red]{13}$$
</s>