Find the points on the graph of the function that are closest to the given point.f(x) = x2,    (0, 6)

Sure, here are the steps to find the points on the graph of the function that are closest to the given point (0,6):

1. The function given is f(x) = x^2. The graph of this function is a parabola.

2. We are looking for the points on this graph that are closest to the point (0,6).

3. To find these points, we need to minimize the distance between the point (0,6) and any point (x, x^2) on the graph of the function.

4. The distance between these two points is given by the distance formula: sqrt((x-0)^2 + (x^2 - 6)^2).

5. To minimize this distance, we can take the derivative of the distance with respect to x, set it equal to zero, and solve for x.

6. The derivative of the distance with respect to x is: 2(x-0) + 2(x^2 - 6)(2x) = 0.

7. Solving this equation for x gives x = 0 and x = sqrt(6).

8. Substituting these values of x into the function f(x) = x^2 gives the points (0,0) and (sqrt(6), 6) on the graph of the function.

9. Therefore, the points on the graph of the function that are closest to the point (0,6) are (0,0) and (sqrt(6), 6).