Differentiation (help please): Find point on curve y=x^2+1 closest to point (18,1)

Farazthesizzler said:

Find the point on the curve y=x²+1 that is closest to the point (18,1).

Please help!!

Click to expand...

Suppose you draw a line from point A (18,1) to the curve. Suppose that line intersects the curve at B (x₁,y₁).

The tangent (to the curve) at B must be perpendicular to the line AB for shortest length of AB.

Sketch the situation and continue......

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Please, solve it with all the steps by taking first derivative

[h=1]Find the point on the curve y = x^2 +1 that is closest to the point (18,1)

I know the distance will be calculated between the imaginary point P(x1,y1) and the point (say A) A(18,1).

For more, please see the image and that’s where I’m stucked. Please solve it further.[/h]

Thanks for indicating what method is required, and showing your work.

You can save a little work by noting that the minimum distance occurs when the square of the distance is minimum, since the square root is a monotonically increasing function. So you really only had to differentiate (x - 18)^2 + x^4.

But evidently you are stuck trying to solve the resulting cubic equation. It can be factored by trying possible linear factors (using the rational root theorem), or by just trying small numbers. Have you done that?