Find the point on the graph of f(x)=√x that is closest to the value (4,0).

(differentiate the square of the distance from a point (x,√x) on the graph of f to the point (4,0).)

Square of distance from (4,0)

D(x)= ((x-4)^2+x)
For minimum, D'(x)=0
2(x-4)+1=0
x=7/2

To find the point on the graph of f(x) = √x that is closest to the value (4,0), we can differentiate the square of the distance between a point (x, √x) on the graph of f and the point (4, 0).

Let's call the distance between these two points d. The formula for the distance between two points in a coordinate plane is:

d = √((x - 4)^2 + (√x - 0)^2)

To simplify this equation, we square both sides to get rid of the square root:

d^2 = (x - 4)^2 + (√x - 0)^2
= (x - 4)^2 + x

Now, we differentiate both sides of the equation with respect to x:

(d^2)' = (x - 4)^2 + x

To differentiate (d^2) with respect to x, we use the chain rule:

(d^2)' = 2d * d'

Since d is the square root of the quantity (x - 4)^2 + x, we can rewrite it as:

d = [(x - 4)^2 + x]^(1/2)

Using the chain rule, we differentiate d:

d' = (1/2) * [(x - 4)^2 + x]^(-1/2) * (2(x - 4) + 1)

Simplifying further, we have:

d' = (x - 4) / [(x - 4)^2 + x]^(1/2)

Substituting d' back into the equation (d^2)' = 2d * d', we have:

2d * [(x - 4) / [(x - 4)^2 + x]^(1/2)] = (x - 4)^2 + x

Next, we can simplify and solve for x:

2d * (x - 4) = [(x - 4)^2 + x]^(3/2)

Simplifying further, we have:

2d * (x - 4) = [(x - 4)^2 + x]^(3/2)

Now, we solve for x by isolating it on one side of the equation. This process involves multiple steps and may require the use of numerical methods, such as iterative approximation or graphing tools like Desmos or WolframAlpha, to find the final answer.

Once we find the value(s) of x that satisfy the equation, we can substitute them back into the original equation f(x) = √x to find the corresponding y-coordinate(s) and obtain the point(s) on the graph of f(x) = √x that are closest to the point (4,0).