Hi! Can I get some help with this questions. I am very confused about how to approach it.

Find the point on the curve y=5x+4 closest to the point (0,6).

Thank you!

y = 5x + 4 is a line, so I am not certain where the curve comes from... did you make a typo?

the closest point will be on a line passing through the given point, and perpendicular to the tangent of the curve

in this case, the curve is a straight line

the slope of the perpendicular is ... -1/5

find the equation of the line with a slope of -1/5, passing through (0,6)

the intersection with the original curve is the closest point

or, if you actually want to use some calculus, the distance d from any point (h,k) to a point on the line y=5x+4 is

d^2 = (h-x)^2 + (k-y)^2
= (h-x)^2 + (k-(5x+4))^2
= h^2-2hx+x^2 - (k^2 - 2(5x+4)k + (5x+4)^2)
= h^2-2hx+x^2 - k^2 - 10kx - 8k - 25x^2-40x-15
= (h^2-k^2) - (2h+10k+40)x - 24x^2
this distance is least when dd/dx=0, or
x = (h+5k+20)/24
or, when h = -5k-20

That is the line with slope -1/5 as shown above.

MsPi: It does says it is a curve that is why I am confused on how to do it.

Scott: What do you mean by the intersection with the original curve is the closest point

Steve: Thank you! And how would I calculate the point from there? I did that process before posting the question here and I couldn't figure out how to get the actual point

Of course, I can help you with that! To find the point on the curve that is closest to the point (0,6), we can use the concept of distance between two points. Here are the steps to approach this problem:

1. Start by writing down the equation of the curve given to us, which is y = 5x + 4.

2. Next, write down the formula for calculating the distance between two points (x1, y1) and (x2, y2). The formula is:

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

In this case, one point is given as (0,6), and we need to find the distance between this point and a point on the curve.

3. Since we know that a point on the curve will have coordinates (x, 5x + 4), we can substitute these values into the distance formula. Thus, the distance formula becomes:

Distance = √[(x - 0)^2 + ((5x + 4) - 6)^2]

4. Simplify the expression inside the square root:

Distance = √[x^2 + (5x - 2)^2]

5. To find the point on the curve closest to (0,6), we need to minimize the distance. In other words, we need to find the value of x that minimizes the expression inside the square root.

6. To find this value, you can differentiate the expression inside the square root with respect to x, set it equal to zero, and solve for x. Alternatively, you can graph the expression and identify the x-coordinate where the graph reaches its minimum.

7. Once you have found the value of x that minimizes the expression inside the square root, substitute this value back into the equation of the curve (y = 5x + 4) to find the corresponding y-coordinate.

8. The point on the curve (y = 5x + 4) closest to (0,6) will have the coordinates (x, y) that you found in the previous step.

I hope this explanation helps you approach the problem! Let me know if you have any further questions.