I want to find the shortest distance between point $(2,0)$ and function $\sqrt{16x^2 + 5x+16}$, So I did the optimization but the answer is always wrong.
$$\begin{align}D=&\sqrt{(x-2)^2+(y)^2}\\ y^2=&{16x^2 + 5x+16}\\ &\text{(by substituting)}\\ D=&\sqrt{(x-2)^2+(16x^2 + 5x+16)}\\ D=& \sqrt{17{x}^{2}+ x + 20}\\ &\text{(Convert to a polynomial since root doesn't matter in the domain)}\\ D(x)=& 17x^2+x+20 \\ D^\prime(x)=& 34x+1 \\ x =& \frac{-1}{34}\end{align}$$
Which is clearly not the closest point, So what did I do wrong?
Here is the question and my answer, which is evaluated as wrong.