To solve this problem, we want to find the point on the curve $y = \sqrt{x}$ that minimizes the distance to the point $(4, 0)$.

Steps to Solve:

1. Distance Formula:
   The distance $D$ between a point on the curve $(x, \sqrt{x})$ and $(4, 0)$ is: $D = \sqrt{(x - 4)^2 + (\sqrt{x} - 0)^2}$ Simplify the distance squared (to avoid working with the square root for now): $D^2 = (x - 4)^2 + (\sqrt{x})^2$ $D^2 = (x - 4)^2 + x$

2. Objective:
   Minimize $D^2$ by finding the critical points of $D^2$.

3. Derivative of $D^2$:
   Let $f(x) = (x - 4)^2 + x$. Differentiate $f(x)$ with respect to $x$: $f'(x) = 2(x - 4) + 1$ $f'(x) = 2x - 8 + 1$ $f'(x) = 2x - 7$

4. Critical Points:
   Solve $f'(x) = 0$: $2x - 7 = 0$ $x = \frac{7}{2} = 3.5$

5. Verify Minimum:
   To confirm that $x = 3.5$ gives a minimum, use the second derivative: $f''(x) = 2 > 0$ Since $f''(x) > 0$, $x = 3.5$ is indeed a minimum.

6. Find the Corresponding Point:
   When $x = 3.5$, $y = \sqrt{3.5}$.
   The point on the curve is: $\left(3.5, \sqrt{3.5}\right)$

Final Answer:

The point on the curve $y = \sqrt{x}$ that minimizes the distance to $(4, 0)$ is: $\boxed{\left(3.5, \sqrt{3.5}\right)}$

Would you like me to plot this, provide further verification, or clarify any step? Here are some related questions to expand your understanding:

1. Why do we minimize $D^2$ instead of $D$?
2. How can we confirm the minimum using numerical methods?
3. How would the solution change if the point was not $(4, 0)$?
4. Can we solve this problem geometrically instead of using calculus?
5. What is the significance of the second derivative test in optimization problems?

Tip: Minimizing the square of the distance avoids unnecessary complexity from square roots, making the derivatives much simpler to handle.