We want to find approximate solutions to the equation x^5 + 3x - 2 = 0 Show that

Question

We want to find approximate solutions to the equation x^5 + 3x - 2 = 0 Show that the function f(x) = x^5 + 3x - 2 has a root between 0 and 1. To use the Newton's method to approximate the root choose an initial value x_0. List x_0,x_1, x_2, x_3, ... until you have found the root correct to 8 decimal places. Use Rolle's Theorem to prose that f(x) has exactly one root. Find a positive number such that the sum of the number and four times its reciprocal is as small as positive. Use your solution to prove that x + 2/x > 3 for all positive numbers x. Imagine a space ship that is stationed on the r-axis at (9, 0) and a comet approaching that station. The trajectory of the comet is found to be parabolic with equation y = 2x^2. Will the comet hit the station? If not, how close will it get to the station?

Explanation / Answer

Since you didn't specified which problem do you want to be solved, and I'm allowed to answer only one problem with four subparts by post, I choose to answer problem number 6.

Here we have a space ship at (9, 0), and a comet with an equation y=2x^2. We know that the comet will not hit the station, since the point (9, 0) does not pass through the given parabola. If we put x=9 in the parabola, we get y=2*9^2=162.
Therefore, we have to calculate the minimum distance that separates the comet with the station.
Let D be this distance, then we have to calculate the distance between the points (9,0) and (x,y) = (x,2x^2). This is given by the formula
D(x) = sqrt{(x-9)^2+2x^2}.
Then, the distance is a function of x. The derivative of this function is
D'(x) = (x-3)/sqrt{x^2/3-2x+9}.
This function clearly has a zero at x=3, which corresponds to a global minimum of the function D(x). Therefore, the minimum distance is reached at the point (3, 2*3^2) = (3, 18), and the distance at this point is 3sqrt{6}.

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