Answer: (4.5, 5.1)
Explanation:
The distance between two points (a, b) and (c, d) is:
D = √( (a - c)^2 + (b - d)^2)
Then if we want to find the point on the curve y = √(5*x + 3) that is closest to the point (7, 0) we need to minimize the distance:
D = √( (x - 7)^2 + (y - 0)^2)
D = √( (x - 7)^2 + (√(5*x + 3))^2)
Because we know that D is positive, minimizing D is the same than minimizing D^2
Then we can minimize:
D^2 = (x - 7)^2 + (5*x + 3)
D^2 = x^2 - 14*x + 21 + 5*x + 3
D^2 = x^2 - 9*x + 24
This is a quadratic equation with a positive leading coefficient, then the minimum of this function is at the vertex.
To find the vertex, we need to find the zero of the first derivative, this is:
(D^2)' = 2*x - 9
We need to solve:
0 = 2*x - 9
9 = 2*x
9/2 = x
4.5 = x
To find the correspondent y-value, we need to evaluate the curve in x = 4.5
y = √(5*4.5 + 3) = 5.1
Then the point is (4.5, 5.1)
This means that the point on the curve y = √(5*x + 3) which is closest to the point (7, 0) is the point (4.5, 5.1)