Final answer:
The minimum distance from the parabola x = y^2 to the point (0,3) is found by using the distance formula and taking the derivative to find the critical point that minimizes the distance. After finding the critical value for y, it's substituted back into the distance formula to get the minimum distance.
Step-by-step explanation:
The student is asking for the minimum distance from the parabola x = y^2 to the point (0,3). To find this, we can use the distance formula between a point and a curve. The distance between any point (x, y) on the parabola and the point (0,3) is given by d = √((x-0)^2 + (y-3)^2) = √((y^2-0)^2 + (y-3)^2) since x = y^2. We can find the minimum distance by finding the derivative of this distance with respect to y, setting that derivative to 0, and solving for y. This gives us a critical point, which we can test to see if it provides a minimum distance.
Let's consider the function d(y) = √(y^4 + (y-3)^2). We want to find the derivative d'(y) and solve for d'(y) = 0. The value of y that minimizes the distance can be substituted back into the distance function to find the minimum distance.
Assuming we find a single critical point and it leads to a minimum (after confirming it's not a maximum or inflection point using second derivative test or other methods), the minimum distance from the parabola to the point (0,3) will be the value of d(y) at that y value.