Find the point (x,y) on the curve y=sqrt(x+4) closest to the origin.
To find the point on the curve that is closest to the origin, we need to minimize the distance between the origin and the point (x, y) on the curve. The distance between two points in a Cartesian plane is given by the distance formula:
Distance = sqrt((x - 0)^2 + (y - 0)^2)
Since the point lies on the curve y = sqrt(x + 4), we can substitute the value of y in terms of x:
Distance = sqrt((x - 0)^2 + (sqrt(x + 4) - 0)^2)
Now, let's simplify this equation:
Distance = sqrt(x^2 + (x + 4))
To minimize the distance, we need to minimize the expression under the square root. Since the square root is a monotonically increasing function, minimizing the expression inside the square root is equivalent to minimizing the expression without the square root. Hence, we can minimize:
x^2 + (x + 4)
To find the minimum, we differentiate the expression with respect to x and set it equal to zero:
d/dx (x^2 + (x + 4)) = 0
2x + 1 = 0
Solving for x, we get:
2x = -1
x = -1/2
Now that we have the x-coordinate, we can substitute it back into the equation y = sqrt(x + 4) to find the corresponding y-coordinate:
y = sqrt((-1/2) + 4)
y = sqrt(7/2)
Therefore, the point (x, y) on the curve y = sqrt(x + 4) closest to the origin is (-1/2, sqrt(7/2)).