Help with graph please- concavity, relative extrema, etc.

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February 6th 2010, 08:40 AM
BigMath

Help with graph please- concavity, relative extrema, etc.

2x^2+1
--------
2x^2-3x

Show any asymptotes, relative extrema, inflection points, concavity, and where the function is increasing or decreasing.

I'm really having trouble with this (Headbang), any help would be GREATLY appreciated.

Thanks guy/girls.
(Rofl)
February 6th 2010, 09:01 AM
VonNemo19

Quote:

Originally Posted by BigMath

2x^2+1
--------
2x^2-3x

Show any asymptotes, relative extrema, inflection points, concavity, and where the function is increasing or decreasing.

I'm really having trouble with this (Headbang), any help would be GREATLY appreciated.

Thanks guy/girls.
(Rofl)

Set $f'(x)=0$ to locate any extrema.

If $f'(x)$ is undefined at some number $x=c$ , then $x=c$ is a vertical asymptote.

If $\lim_{x\to\pm\infty}f(x)=L$ , then $f(x)=L$ is a horizontal asymptote.

Get started with that and then show me what you've got.
February 6th 2010, 09:05 AM
BigMath

That's basically all I know how to do:

For H.A. I got: 1

For V.A. I got: 3/2, 0

For the increasing/decreasing do i solve for first derivative than make it equal 0? Or do I use the V.A. points and plug in to the first derivative.
February 6th 2010, 09:11 AM
VonNemo19

Quote:

Originally Posted by BigMath

For the increasing/decreasing do i solve for first derivative than make it equal 0?

$f'(x)>0$ means that $f$ is increasing.

$f'(x)<0$ means that $f$ is decreasing.

So, you must solve these inequalities to determine the invervals over which $f$ is increasing or decreasing.
February 6th 2010, 09:18 AM
BigMath

My teacher wanted us to make a chart, with infinity to certain number, certain number to another, and that number to negative infinity and to plug in numbers to get increasing/decreasing. Can you show me?
February 6th 2010, 09:19 AM
VonNemo19

Quote:

Originally Posted by BigMath

That's basically all I know how to do:

For H.A. I got: 1 correct

For V.A. I got: 3/2, 0 correct

For the increasing/decreasing do i solve for first derivative than make it equal 0? Or do I use the V.A. points and plug in to the first derivative.

.
February 6th 2010, 09:25 AM
BigMath

For the numbers you plug in the first derivative, to see if they are greater or less than 0, where do you get those numbers from? Do you use the V.A.?
February 6th 2010, 09:28 AM
BigMath

Are they

-infinity, -1.115

-1.115, 0.448

0.448, 0

0, 3/2

3/2, infinity

Than plug in numbers in between to see increasing or decreasing?
February 6th 2010, 09:41 AM
VonNemo19

Quote:

Originally Posted by BigMath

My teacher wanted us to make a chart, with infinity to certain number, certain number to another, and that number to negative infinity and to plug in numbers to get increasing/decreasing. Can you show me?

I'm not skilled enough with LaTeX to make a chart on this forum, but the idea is to choose test values between the critical numbers to see if $f'>0$ or $f'<0$ .

$f(x)=\frac{2x^2+1}{2x^3-3x}$

$f'(x)=\frac{(2x^2-3x)(4x)-(2x^2+1)(6x^2-3)}{(2x^3-3x)^2}$

$=\frac{(8x^3-12x^2)-(12x^4-3)}{(2x^3-3x)^2}$

$=\frac{(8x^3-12x^2)-(12x^4-3)}{(2x^3-3x)^2}=0$ implies

$(8x^3-12x^2)-(12x^4-3)=0$

$8x^3-12x^2-12x^4+3=0$

$12x^4-8x^3+12x^2-3=0$ Which is really ugly. Are you sure that this is the right problem?
February 6th 2010, 09:46 AM
BigMath

http://www.mathhelpforum.com/math-he...e0ff24fe-1.gif

The bottom is 2x^2-3x

Derivative comes out to like:

-6x^2-4x+3
----------
(2x^2-3x)^2

No idea why that is underlined

So when I solve for 0, I get

-6x^2-4x+3=0

I used the quadratic formula.
February 6th 2010, 09:54 AM
VonNemo19

Quote:

Originally Posted by BigMath

http://www.mathhelpforum.com/math-he...e0ff24fe-1.gif

The bottom is 2x^2-3x

Derivative comes out to like:

-6x^2-4x+3
----------
(2x^2-3x)^2

No idea why that is underlined

So when I solve for 0, I get

-6x^2-4x+3=0

I used the quadratic formula.

ooops. My bad. OK, yeah, so it doesn't factor. Using the quadratic formula should give the two answers that you've given before.

Yeah. You're on the right track. So, make a table that describes the sign of f prime at some test value on the intervals that you have already discovered a couple of posts ago before I went in left field with doing the wrong problem.
February 6th 2010, 09:59 AM
BigMath

Okay I got this:

Than I plug it into the first derivative to see increasing or decreasing:

f1(x) f(x)
-infinity, -1.115 - dec

-1.115, 0.448 + inc

0.448, 0 + inc

0, 3/2 - dec

3/2, infinity - dec

Are those the right points I would use? So can you help me check this so far:

Increasing/decreasing: see above

Relative max: (0, f(0))

Relative min: (-1.115, f(-1.115))

Right or wrong?