Question

How do you find intervals of monotonicity and local extrema of functions?
f(x)= x^3-x

Answer 1

right find the derivative of f(x)

Answer 2

then set it equal to zero

Answer 3

and u will get extrema and test it whether the value is increasing or decreasing if f'(x)= 0 then the test fail

Answer 4

aight u not even paying attention lol bye :D

Answer 5

I'm sorry i'm new to this and didn't realize I had a response.

Answer 6

"Local extrema" would include relative minima and relative maxima. (Forget about "absolute min. and max" for now.

Please take the derivative of the given function and set the derivative = to 0. Show your work. What results do you obtain?

Answer 7

How do I show you my work?

Answer 8

Must be interesting exploring OpenStudy, given that you're new. Need your attention, tho.'

Type it in: for example, f(x)=x^3-x, or \[f(x)=x^3-x\]

Or draw it:

Answer 9

Or take a screen shot and up load the image to OpenStudy.

Answer 10

Okay sorry still trying to figure this out. Working on the problem now.

Answer 11

so it would be 3x^2-1 for the derivative?

Answer 12

3x^2-1=0
3x^2=1
x^2=(1/3)
x= square root of (1/3)?

Answer 13

Yes, but there are actually TWO roots. One is \[+\sqrt{\frac{ 1 }{ 3 }}.\] The other is \[-\sqrt{\frac{ 1 }{ 3}}.\]

Answer 14

Please find the associated y value for each, using the original function.

Answer 15

Why both positive and negative?

Answer 16

First, you could square both of those results. In both cases you'd get +1/3.

Secondly, an nth order polynomial has n roots.

Your f '(x)=x^2-1 is a 2nd order poly. and thus has 2 roots.

f(x)=x^3-x is a third order poly and has 3 roots.

And so on.

Answer 17

Given those 2 roots of f '(x) = 0, find the associated y-values, please.

Answer 18

Oh okay you use the exponent? Okay one second please.

Answer 19

The function is f(x)=x^3-x.
Evaluate this at x=Sqrt(1/3). Your result? You might want to draw your result; it's a bit tricky to type it in.

Answer 20

Here is what I'd type in myself:\[f(\frac{ 1 }{ \sqrt{3} })=(\frac{ 1 }{ \sqrt{3} })^3-\frac{ 1 }{ \sqrt{3} }\]

Answer 21

This would be the value of the function at x=(1/Sqrt(3)).

Answer 22

The two points you obtain this way are your local (or relative) extreme values.