Find the point on the graph of the function closest to the given point. Function f(x) = xsquared Point (2, 1/2)

using distance from (2,1/2) to y=x^2,

d^2 = (x-2)^2 + (y-1/2)^2
= (x-2)^2 + (x^2 - 1/2)^2
= x^2 - 4x + 4 + x^4 - x^2 + 1/4
= x^4 - 4x + 17/4

d = sqrt(x^4 - 4x + 17/4)
d' = 2(x^3 - 1)/sqrt(x^4 - 4x + 17/4)
d' = 0 when x = 1

so, (2,1/2) is closest to (1,1)
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using normal line, we want the line from (2,1/2) normal to the curve. The distance from P to the curve will be minimum along the line normal to the curve.

At (x,y) the slope of x^2 = 2x
so, the normal has slope -1/2x

So, the equation of the normal line from (2,1/2) is

(y - 1/2)/(x-2) = -1/2x

2x(x^2 - 1/2) = -(x-2)
2x^3 - x = -x + 2
2x^3 = 2
x = 1

So, the normal line from (2,1/2) is the line to (1,1)

Well, let's see here. The function f(x) = x^2 is a simple parabola. Now, the point (2, 1/2) is located at the bottom of the parabola. So, if we draw a vertical line from the point (2, 1/2) to the graph of the function f(x) = x^2, it will hit the graph at two points.

Since we're looking for the closest point, we need to find the point on the graph that is equidistant from the point (2, 1/2). Now, if we take a look at the graph, it's clear that the point closest to (2, 1/2) is actually (2, 4)!

Haha! Bet you didn't see that one coming. It turns out that the point (2, 1/2) is actually closer to the vertex of the parabola, which is at (2, 0). So the closest point on the graph to (2, 1/2) is (2, 0). Keep your distance!

To find the point on the graph of the function f(x) = x^2 closest to the point (2, 1/2), we can use the distance formula.

The distance between two points (x1, y1) and (x2, y2) is given by the formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, we have the point (2, 1/2) as (x1, y1), and we want to find the point on the graph of f(x) = x^2 that is closest to it.

To find the point on the graph of f(x) = x^2, we need to determine the coordinates (x, f(x)). Since f(x) = x^2, the coordinates of any point on the graph will be (x, x^2).

Now, let's substitute the values into the distance formula using (x1, y1) as (2, 1/2):

Distance = sqrt((x - 2)^2 + (x^2 - 1/2)^2)

We want to find the value of x that minimizes the distance between the two points. To do this, we need to find the derivative of the distance equation and set it equal to zero. However, solving this equation can be quite complicated.

Alternatively, we can use a graphing calculator or software to plot the graphs of the distance equation and the function f(x) = x^2. By analyzing the graph, we can find the point on the graph of f(x) = x^2 that is closest to (2, 1/2).

Using a graphing calculator, you can plot the graphs of both equations and find the point of intersection closest to (2, 1/2).

To find the point on the graph of the function closest to the given point (2, 1/2), we can use the distance formula. The distance between two points in a coordinate plane is given by the formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, the given point is (2, 1/2) and the graph of the function is defined by f(x) = x^2. To find the point on the graph closest to the given point, we need to find the value of x that minimizes the distance between (x, x^2) and (2, 1/2).

Let's break down the steps to find the closest point:

1. Write the distance formula using the coordinates of the two points:
Distance = sqrt((x - 2)^2 + (x^2 - 1/2)^2)

2. Square both sides of the equation to eliminate the square root:
Distance^2 = (x - 2)^2 + (x^2 - 1/2)^2

3. Expand and simplify the equation:
Distance^2 = (x^2 - 4x + 4) + (x^4 - x^2 + x - 1/4)

4. Collect like terms and simplify the equation:
Distance^2 = x^4 - x^2 + x - 1/4 - 4x + 4
Distance^2 = x^4 - x^2 - 4x + (4 - 1/4)

5. Combine the constants:
Distance^2 = x^4 - x^2 - 4x + 15/4

6. To minimize the distance, we take the derivative of the equation with respect to x and set it equal to zero:
d/dx (Distance^2) = 0

7. Find the derivative of the equation and simplify:
d/dx (x^4 - x^2 - 4x + 15/4) = 0
4x^3 - 2x - 4 = 0

8. Solve for x by factoring or by using numerical methods such as the Newton-Raphson method.

By finding the value of x that solves the equation 4x^3 - 2x - 4 = 0, we can determine the x-coordinate of the point on the graph of the function closest to the given point.

Once we find this x-coordinate, we can substitute it back into the equation f(x) = x^2 to obtain the y-coordinate of the closest point.