How do I find the points on the graph of the function f(x) = x^2 that are closest to the point (0, 4)?

To find the points on the graph of the function f(x) = x² that are closest to the point (0, 4), we can follow these steps:

Identify the Distance Formula: We will use the distance formula to calculate the distance between any point (x, f(x)) on the graph and the point (0, 4). The distance d can be defined as:

d = √[(x – 0)² + (f(x) – 4)²]

Substitute f(x): Since f(x) = x², we can substitute this into the distance formula:

d = √[x² + (x² – 4)²]

Minimize the Distance: To find the closest points, we can minimize d. Instead of minimizing d directly, we can minimize d² to avoid dealing with the square root:

d² = x² + (x² – 4)² = x² + (x^4 – 8x² + 16)

d² = x^4 – 7x² + 16

Differentiate: Next, we take the derivative of d² with respect to x and set it to 0 to find the critical points:

f'(x) = 4x³ – 14x

Setting f'(x) = 0 gives us:

4x³ – 14x = 0

4x(x² – 3.5) = 0

This results in the points:

Evaluate Points: Now we need to find the corresponding f(x) values at these x values:

For x = 0: f(0) = 0² = 0 → point (0, 0)

For x = √(3.5): f(√(3.5)) = (√(3.5))² = 3.5 → point (√(3.5), 3.5)

For x = -√(3.5): f(-√(3.5)) = (-√(3.5))² = 3.5 → point (-√(3.5), 3.5)

Closest Points: Now, we have three potential points on the graph: (0, 0), (√(3.5), 3.5), and (-√(3.5), 3.5). To determine which point is closest to (0, 4), we can calculate the distances:

Distance to (0, 4):

From (0, 0): d = √[(0 – 0)² + (0 – 4)²] = 4
From (√(3.5), 3.5): d = √[(√(3.5) – 0)² + (3.5 – 4)²] = √[3.5 + 0.25] = √[3.75] ≈ 1.936
From (-√(3.5), 3.5): d = √[(-√(3.5) – 0)² + (3.5 – 4)²] = √[3.5 + 0.25] = √[3.75] ≈ 1.936

Comparatively, the closest points on the graph of the function f(x) = x² to the point (0, 4) are:

Both points at approximately (1.87, 3.5) and (-1.87, 3.5) are the closest to the point (0, 4) on the graph.