Find the points on the graph of the function that are closest to the given point.

*f*(*x*) = *x*2,

    

(0, 9)

smaller *x*-value

(*x*, *y*)

=
 
 
  
larger *x*-value

(*x*, *y*)

=
 
 
  

We are tasked with finding the points on the graph of the function \( f(x) = x^2 \) that are closest to the point \( (0, 9) \). Here's the step-by-step solution to solve this optimization problem.

### Step 1: Define the Distance Formula
The distance between a point \( (x, y) \) on the curve and the given point \( (0, 9) \) is given by the distance formula:

\[
d = \sqrt{(x - 0)^2 + (y - 9)^2}
\]

Since \( y = x^2 \) (from the equation of the curve), we substitute this into the distance formula:

\[
d = \sqrt{x^2 + (x^2 - 9)^2}
\]

This simplifies to:

\[
d = \sqrt{x^2 + (x^4 - 18x^2 + 81)}
\]

\[
d = \sqrt{x^4 - 17x^2 + 81}
\]

### Step 2: Minimize the Distance
To minimize the distance \( d \), we can minimize \( d^2 \), as it simplifies the process. Let:

\[
D(x) = x^4 - 17x^2 + 81
\]

We will now minimize \( D(x) \).

### Step 3: Take the Derivative of \( D(x) \)
Differentiate \( D(x) \) with respect to \( x \):

\[
D'(x) = 4x^3 - 34x
\]

### Step 4: Set the Derivative Equal to Zero
Set \( D'(x) = 0 \) to find the critical points:

\[
4x^3 - 34x = 0
\]

Factor the equation:

\[
2x(2x^2 - 17) = 0
\]

This gives us two solutions:

1. \( x = 0 \)
2. \( 2x^2 - 17 = 0 \)

Solve for \( x^2 \) in the second equation:

\[
x^2 = \frac{17}{2}
\]

Thus, \( x = \pm \sqrt{\frac{17}{2}} = \pm \frac{\sqrt{34}}{2} \).

### Step 5: Find the Corresponding \( y \)-values
For \( x = 0 \):

\[
y = f(0) = 0^2 = 0
\]

For \( x = \pm \frac{\sqrt{34}}{2} \):

\[
y = f\left(\pm \frac{\sqrt{34}}{2}\right) = \left(\pm \frac{\sqrt{34}}{2}\right)^2 = \frac{34}{4} = \frac{17}{2}
\]

### Step 6: Points on the Graph
- For the smaller \( x \)-value \( x = -\frac{\sqrt{34}}{2} \), the point is \( \left(-\frac{\sqrt{34}}{2}, \frac{17}{2}\right) \).
- For the larger \( x \)-value \( x = \frac{\sqrt{34}}{2} \), the point is \( \left(\frac{\sqrt{34}}{2}, \frac{17}{2}\right) \).

Thus, the points on the graph closest to \( (0, 9) \) are:

- Smaller \( x \)-value: \( \left(-\frac{\sqrt{34}}{2}, \frac{17}{2}\right) \)
- Larger \( x \)-value: \( \left(\frac{\sqrt{34}}{2}, \frac{17}{2}\right) \)

Would you like further details or have any questions?

### 5 Relative Questions:
1. How do we derive the critical points when minimizing a function?
2. Can you explain why minimizing \( d^2 \) is easier than minimizing \( d \)?
3. What happens when a point on the graph lies directly above or below the given point?
4. How does the distance formula change if the point moves along the y-axis?
5. Can we always assume symmetry when solving optimization problems on parabolas?

### Tip:
When dealing with distance minimization problems, always simplify the square of the distance to avoid working with square roots until the final step.

Find the points on the graph of the function that are closest to the given point. f(x) = x^2, (0, 9), smaller x-value (x, y) = larger x-value (x, y) =

This step-by-step guide solves an optimization problem to find points on the parabola f(x) = x² that are closest to the point (0, 9). Learn about distance minimization, derivatives, and solving critical points. Suitable for Grades 11-12 students.

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