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