A track has a height that is a function of horizontal position x, given by h(x) = x^3 + 4x^2 − 44x + 16.

Question

A track has a height that is a function of horizontal position x, given by h(x) = x^3 + 4x^2 − 44x + 16.

Find all the positions on the track where a marble will remain where it is placed. What kind of equilibrium exists at each of these positions? (Enter your answers from smallest to largest.)

The answers are -5.39 and 2.72. I'm not sure where to start this problem. I thought that I would have to take h(x)'s derivative and set it equal to zero, but it did not work. Any help is appreciated.

Answer 1

So you want to find the minimum heights...

h'=0=3x^2+8x-44

use the quadratic equation to solve for the x positions.

x=(-8+-sqrt(64+4*3*44))/6=-4/3 +-4.05

which is your answers.

Answer 2

Thank you. I completely forgot about the quadratic formula.

Answer 3

The quadratic equation is not really working. I keep getting imaginary numbers from the negative under the radical.

Answer 4

Under the radical, you should have (as bobpursley had it) 64-(4)(3)(-44)=592

Answer 5

I forgot about the negative. Thank you!

Answer 6

A track has a height that is a function of horizontal position x, given by h(x) = x^3 + 4x^2 − 44x + 16.

So you want to find the minimum heights...

Thank you. I completely forgot about the quadratic formula.

The quadratic equation is not really working. I keep getting imaginary numbers from the negative under the radical.

Under the radical, you should have (as bobpursley had it) 64-(4)(3)(-44)=592

I forgot about the negative. Thank you!

Thank you for the help. I managed to get the correct solution.

Great!