Find the point (x,) on the graph of the curve y=√4x+13 that is the closest to the fixed point (5,0).

EDIT: Looking for the point (x,y) not (x,). My apologies for the typo.

is the equation

y = √(4x + 13) , or
y = √(4x) + 13

I will assume y = √(4x+13)

let the point of contact be P(x,y)
or P(x, √(4x+13) )

let D be the distance from P to (5,0)
D = (√(4x+13) - 0)^2 + (5-x)^2 )
= √( (4x+13) + (5-x)^2 )
square both sides to make it easier to differentiate
D^2 = 4x+13 + (5-x)^2
= 4x + 13 + 25 - 10x + x^2
= x^2 - 6x + 38

2D dD/dx = 2x - 6

for a max/min of D , dD/dx = 0

2x - 6 = 0
x = 3
so P is (3, √25) or P(3,5)

just noticed that in my

D = (√(4x+13) - 0)^2 + (5-x)^2 )
it should have been
D = (√(4x+13) - 0)^2 + (x-5)^2 )
as well as the following 2 lines.

but since we squared it, that error did not matter.

To find the point (x, y) on the graph of the curve that is closest to the fixed point (5, 0), we need to minimize the distance between these two points.

The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

In our case, we have the fixed point (5, 0) and any point on the curve (x, y) that satisfies the equation y = √(4x + 13).

So, the distance formula becomes:

Distance = √((x - 5)^2 + (y - 0)^2)

Substituting y = √(4x + 13) into the distance formula:

Distance = √((x - 5)^2 + (√(4x + 13) - 0)^2)

Now, we want to minimize this distance. To do that, we can take the derivative of the distance function with respect to x and set it equal to 0 to find the critical points.

Differentiating the distance function:

d(Distance)/dx = (1/2) * ( 2(x - 5) + 2(√(4x + 13) * (1/2) * (4/(2√(4x + 13))) )

Simplifying:

d(Distance)/dx = (x - 5) + (2(2x + 13)) / (2√(4x + 13))

Setting the derivative equal to 0:

(x - 5) + (2(2x + 13)) / (2√(4x + 13)) = 0

Now we can solve this equation to find the critical point(s) that gives us the minimum distance.

Simplifying and solving for x (critical point):

(x - 5) + (2(2x + 13)) / (2√(4x + 13)) = 0

Multiply both sides by 2√(4x + 13):

2(x - 5)√(4x + 13) + 2(2x + 13) = 0

Simplifying the equation:

2(x - 5)√(4x + 13) = -2(2x + 13)

Dividing both sides by -2:

(x - 5)√(4x + 13) = x + 13

Squaring both sides to eliminate the radical:

(x - 5)^2(4x + 13) = (x + 13)^2

Expanding and simplifying:

4x^3 - 26x^2 - 5x + 52 = 0

At this point, it is difficult to solve this cubic equation analytically. You can use numerical methods like the Newton-Raphson method or graphing calculators/software to find the approximate values of x (which gives the x-coordinate of the closest point).

Once you have the x-coordinate, you can substitute it back into the equation y = √(4x + 13) to find the corresponding y-coordinate of the closest point.

Alternatively, you can plot the graph of y = √(4x + 13) and the point (5, 0) on a graphing calculator or software. The point on the graph closest to (5, 0) will be visually apparent as the point that is nearest to the fixed point.