https://file.aiquickdraw.com/mathbot/file/17270629387752agvja8g.jpg

The problem in the image involves the graph of \( y = \sqrt{x} \) and asks about the distance from a point \( P = (x, y) \) on this graph to a fixed point \( (4, 0) \). Let's break down each part:

### Question Breakdown:
- **(a)** Find the distance from \( P(x, \sqrt{x}) \) to the point \( (4,0) \) as a function of \( x \).
- **(b)** Use a graphing utility to graph the function \( d(x) \).
- **(c)** Determine the value(s) of \( x \) that minimize the distance \( d(x) \).
- **(d)** Find the smallest distance.

### Step-by-Step Solution:

#### (a) Express the distance function \( d(x) \)
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the plane is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Here, \( P = (x, \sqrt{x}) \) and \( (4, 0) \). Substituting these into the distance formula:

\[
d(x) = \sqrt{(4 - x)^2 + (\sqrt{x} - 0)^2}
\]
Simplify:

\[
d(x) = \sqrt{(4 - x)^2 + (\sqrt{x})^2} = \sqrt{(4 - x)^2 + x}
\]

So, the distance function is:

\[
d(x) = \sqrt{(4 - x)^2 + x}
\]

#### (b) Graph \( d(x) \)
You can use any graphing utility (like Desmos, GeoGebra, or a graphing calculator) to graph the function \( d(x) = \sqrt{(4 - x)^2 + x} \). It will show how the distance changes as \( x \) varies.

#### (c) Find the value(s) of \( x \) where \( d(x) \) is minimized
To minimize the distance, you would typically differentiate \( d(x) \) with respect to \( x \) and find the critical points. 

1. Differentiate \( d(x) \):

\[
d'(x) = \frac{d}{dx} \left( \sqrt{(4 - x)^2 + x} \right)
\]

To find the critical points, solve \( d'(x) = 0 \).

#### (d) What is the smallest distance?
Once you have the value of \( x \) that minimizes the distance, plug that value back into \( d(x) \) to find the smallest distance.

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Would you like a detailed calculation of the derivative and solution for the minimum, or help with graphing it?

### 5 Related Questions:
1. How do you compute the derivative of \( d(x) = \sqrt{(4 - x)^2 + x} \)?
2. What does the graph of \( d(x) \) reveal about the behavior of the distance as \( x \) changes?
3. What is the significance of finding the critical points of a function?
4. How can we verify that the critical point found minimizes the function \( d(x) \)?
5. What real-world applications can involve minimizing the distance between a point and a curve?

### Tip:
When solving optimization problems, always verify if the critical points found are minima by checking the second derivative or analyzing the graph.

Let P = (x, y) be a point on the graph of y = √x. (a) Express the distance d from P to the point (4, 0) as a function of x. (b) Use a graphing utility to graph d = d(x). (c) For what value(s) of x is d smallest? (d) What is the smallest distance?

Learn how to express and graph the distance function from a point on the curve y = √x to a fixed point (4,0). Understand how to find the value of x that minimizes this distance using algebra and calculus techniques. Suitable for advanced high school students in Grades 11-12.

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