Question

Consider the function f(x)= 3/4x^4-x^3-3x^2+6x
Find the relative extrema for f(x); be sure to label each as a maximum or minimum. You do not need to find function values; just find the x-values.
Determine the interval(s) where f(x) is increasing (if any) and the interval(s) where f(x) is decreasing (if any).

3. Show why there are exactly two inflection points for this function. Note: You do NOT need to find the inflection points.

4.. Show f(x) is concave up at x = −2, x = −1, and x = 2. And show f(x) is concave down at x = 0. Use this information and the information in parts A–C to sketch this function. (How would I sketch this?)

Answer 1

For greater clarity, I'm going to use Equation Editor to present this function:\[f(x)= \frac{ 3 }{ 4 } x^4-x^3-3x^2+6x \] (1) Please find the first derivative of this.
(2) Set this first derivative = to 0 and solve for the critical values (x-values at which f '(x) is zero).

Answer 2

It's important that you go through that process (of finding the zeros).

But you may find it helpful to see a graph of this function to help you locate the relative minimum and the horizontal point of inflection:

http://www.wolframalpha.com/input/?i=f%28x%29%3D+%283%2F4%29x%5E4-x%5E3-3x%5E2%2B6x+

Answer 3

3(x^3 - x^2 -2x+2)?

Answer 4

0= 3(x^3 - x^2 - 2x+2)

Answer 5

Very nice. Would you please set this equal to zero and determine any real roots of that equation? How would you find such roots?

Answer 6

What course are you in? That would partially determine which method you should use to find the real roots of this equation.

Answer 7

To find the real root we split the equation and set each part to 0?

Answer 8

I am in Calc

Answer 9

something like 3= 0 and x^3- x^2-2x+2 =0

Answer 10

You're talking about factoring. Your factors would possilbly not contain whole numbers, according to the graph I've shared with you.

What level Calculus is your course? Have you heard of and/or used Newton's Method?

Answer 11

AP calc, but I don't think it was defined as Newton's method

Answer 12

Was the term "numerical methods" used?

Answer 13

Hmm I think so... but I think so

Answer 14

But I more so remember how to do it a little and not so much the name

Answer 15

I'm going to go through our conversation once more to ensure that there are no oversights or mistakes.
Supposing that you and I had not started this conversation. What approach or approaches might you have used to find the critical values of f(x)= 3/4x^4-x^3-3x^2+6x?

Answer 16

Well I when I first tried to solve i 1. found the derivative 2. set each part to zero (3= and then x^3 - x^2 - 2x + 2 3. I factored the derivative...

Answer 17

Have you seen this formula before? It's Newton's Method:\[x _{2}=x _{1}-\frac{ f(x _{1}) }{ f '(x _{1}) }\]

Answer 18

Factoring would work if you were either very lucky or very smart in identifying the roots.
What are your factors? What are your roots?
Have you seen that formula before (see immediately above)?

Answer 19

Hmm I think so yes but I forget the equations because my teacher doesn't exactly like teaching equations rather than the process
, but I sort of remember the process

Answer 20

all right. Let's think of what we could do at this point that would most help you. What info or guidance would you like to receive from me at this point?

Answer 21

For whatever it's worth, I've determined that x=Sqrt(2) is a root of the derivative, which means that (x-Sqrt(2) ) is a factor of the derivative. does that agree at all with one of your roots / your factors?

Answer 22

Well I am pretty sure I was doing it correctly because my teacher told me my method seemed to be accurate. But I am not certain.

Answer 23

OK. We can take this discussion as far as you wish, or we could stop now, or anything in between. Have you any specific questions?

Answer 24

well I messed up well I am not sure. I first wrote 3(x^3 - x^2 -2x+2) = 0 3=0 x^2-2x+2 =0
then I think it's
3=0
(x-1)=0= -1
(x^2 -2)=0 = -2

Answer 25

It's always better to be able to find roots of a polynomial yourself if you possibly can.

However, as a back-up, or as a means of checking your results, y ou could use wolframalpha.com, typing in the 1st derivative in the entry box; this software will graph the function for y ou and indicate what the roots are. At least that'd give you a way of verifying the correctness of y our answers.

See: http://www.wolframalpha.com/input/?i=+x%5E3+-+x%5E2+-+2x+%2B+2

I need to get off the Internet now, but will look for your replies when I do. Great working with you; I hope to have the pleasure again.

Answer 26

thank you