To find the point on the graph of $f(x) = \sqrt{x - 7}$ that is closest to the point $(10, 0)$, we can proceed with the following steps:

1. Define the Distance Function:
   The distance $d$ between a point on the curve $(x, f(x))$ and the point $(10, 0)$ is given by: $d = \sqrt{(x - 10)^2 + \left( \sqrt{x - 7} - 0 \right)^2} = \sqrt{(x - 10)^2 + (x - 7)}$

2. Simplify the Distance Function:
   To make it easier to work with, we'll minimize the square of the distance instead of $d$ itself (since the square root is a monotonic function, this won't change the minimum point): $d^2 = (x - 10)^2 + (x - 7)$ Expanding this expression: $d^2 = (x - 10)^2 + x - 7 = x^2 - 20x + 100 + x - 7 = x^2 - 19x + 93$

3. Find the Minimum of $d^2$:
   To minimize $d^2$, we take the derivative with respect to $x$ and set it to zero: $\frac{d}{dx}(d^2) = 2x - 19 = 0$ Solving for $x$: $x = \frac{19}{2} = 9.5$

4. Find $y$ Value on the Curve:
   Now, substitute $x = 9.5$ into $f(x)$ to find $y$: $f(9.5) = \sqrt{9.5 - 7} = \sqrt{2.5}$

So, the point on the graph closest to $(10, 0)$ is: $(x, y) = \left( 9.5, \sqrt{2.5} \right)$

Would you like a more detailed explanation, or have any questions?

Related Questions:

1. How do you find the minimum distance from a point to a line instead of a curve?
2. Why does minimizing $d^2$ yield the same result as minimizing $d$?
3. Can the method used here be applied to other functions, like polynomial functions?
4. How would the process change if the point were further away, say at $(20, 0)$?
5. How can we use calculus to verify that our minimum distance is indeed the closest point?

Tip:

In optimization problems involving distance, minimizing the square of the distance often simplifies calculations by removing the square root, while still preserving the location of the minimum.