To find the point on the graph of the function closest to the given point (9, 0), we need to minimize the distance between the two points. Let's start by finding the distance formula. Then we can find the point by differentiating the expression for the distance and setting it equal to zero.Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)In our case, (x1, y1) = (9, 0) and (x2, y2) = (x, sqrt(x)). Plugging these values into the distance formula, we get:Distance = sqrt((x - 9)^2 + (sqrt(x) - 0)^2)To minimize the distance, we can minimize the square of the distance. The squared distance is:Distance^2 = (x - 9)^2 + (sqrt(x))^2To find the minimum value, we can take the derivative of the squared distance with respect to x and set it equal to zero. Differentiating the expression, we get:d(Distance^2)/dx = 2(x - 9) + 2 * sqrt(x) * (1/2) * (1 / sqrt(x))Simplifying this expression, we have:2(x - 9) + 1 = 02x - 18 + 1 = 02x - 17 = 02x = 17x = 17/2So the x-coordinate of the point on the graph closest to (9, 0) is 17/2. To find the y-coordinate, we can substitute this value back into the function f(x) = sqrt(x):f(17/2) = sqrt(17/2)Simplifying this, we get:f(17/2) = sqrt(17/2) * sqrt(2/2)f(17/2) = sqrt(34/4)f(17/2) = sqrt(34)/sqrt(4)f(17/2) = sqrt(34)/2Therefore, the point on the graph that is closest to (9, 0) is (17/2, sqrt(34)/2).
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