Find the point on the curve `y = 3x^2 + 4x + 2` that is closest to the point `(2, 4)`

Expert Answers
txmedteach eNotes educator| Certified Educator

To answer this question, we need to determine the distance between the point and the curve. We then need to minimize that distance.

Let's start with the distance formula:

`d = sqrt((y_2-y_1)^2 + (x_2-x_1)^2)`

For simplicity, let `(x_2,y_2)` be on the curve, and let `(x_1,y_1)` be the given point:

`(x_1,y_1) = (2,4)`

`(x_2,y_2) = (x, 3x^2+4x+2)`

Let's now substitute these values into the distance formula:

`d = sqrt((3x^2+4x+2-4)^2 + (x-2)^2)`

Now, let's simplify, first in the parentheses, then by distribution:

`d = sqrt((3x^2+4x-2)^2 + (x-2)^2)`

`d = sqrt(9x^4 + 24x^3 - 12x^2 + 16x^2 - 16x +4 + x^2 - 4x + 4)`

`d = sqrt(9x^4 + 24x^3+5x^2-20x+8)`

Now that we have our distance, we need to minimize it. To do this, we will take the derivative with respect to x and set that derivative to zero. We make sure to use the chain rule!

`(dd)/(dx) =(36x^3 + 72x^2 + 10x - 20)/sqrt(9x^4+24x^3+5x^2-20x+8)`

Now, set the derivative to zero:

`0 = (36x^3 + 72x^2 + 10x - 20)/sqrt(9x^4+24x^3+5x^2-20x+8)`

We can multiply both sides by the denominator with no consequence:

`0 = 36x^3 + 72x^2 + 10x - 20`

We now find the roots of this equation. Without any good way to do this algebraically, let's just go ahead and find the roots in a calculator. If you use the plotting function, you'll notice that there are 3 possibe roots: x = 0.425, x = -0.810, x = -1.615.

Let's examine the graph of the distance function to determine which of these three x-values will give us the minimum distance.

Clearly, the distance value at x = 0.425 is the minimum value. So, let's evaluate this to give us our minimum distance:

`d = sqrt(9*0.425^4 + 24*0.425^3 + 5*0.425^2 - 20*0.425 + 8) = 1.59`

The question asks us to find the point on the curve at which this minimum distance is achieved, so let's input our x-value into the equation for our final result:

`y = 3x^2 + 4x + 2 = 3(0.425)^2 + 4(0.425)+2 = 4.24`

The closest point on the curve to the point `(2,4)` will be `(0.425, 4.24).`

To confirm, let's look at a graph of the curve and point:

Looks like we have the correct point! I hope this helps!