What is the maximum number of relative extrema contained in the graph of this function? I'm getting 0? Is that right?
f(x)=3x^4-x^2+4x-2

Question

What is the maximum number of relative extrema contained in the graph of this function? I'm getting 0? Is that right?
f(x)=3x^4-x^2+4x-2

myininaya · Answer

You found f' correct?

myininaya · Answer

If so what did you get?

anonymous · Answer

Im still not for sure on how to solve the problem in general. I'm solving it off a video I watched. I know you set it equal to zero, and thats what I did, but after that I'm very fuzzy on what exactly to do

anonymous · Answer

You find the f'(x) (derivative) of f(x) and set it equal to zero, then solve for x.

anonymous · Answer

That's the first step.

myininaya · Answer

do you know power rule for differentiating?

myininaya · Answer

$$(x^4)'=? \ (x^2)'=? $$

anonymous · Answer

This is what I got for the derivative? is that correct?$$12x^3-2x+4$$

anonymous · Answer

Yes.

myininaya · Answer

not set =0 and solve for x
but eww yuck a cubic

anonymous · Answer

So x would be -0.77317?

anonymous · Answer

That's the only solution to that equation yes, but you can't determine it's the relative extrema just from that.

anonymous · Answer

Okay so then what would I do next to determine the relative extrema?

myininaya · Answer

http://www.wolframalpha.com/input/?i=12x%5E3-2x%2B4%3D0 yep I would find f'' also and plug tht value into f''

myininaya · Answer

f''(a)>0 implies at x=a there is a min
f''(a)<0 implies at x=a there is a max

anonymous · Answer

Could you possibly write it out and show me how to do that?

myininaya · Answer

to find f'' you need to find the derivative of f'

myininaya · Answer

can you do that?

myininaya · Answer

$$f=3x^4-x^2+4x-2 \ f'=12x^3-2x+4 \ f''=(f')'=(12x^3-2x+4)'=?$$

anonymous · Answer

$$36x^2-2$$ Is that it?

myininaya · Answer

yep

myininaya · Answer

now replace x with the critical value you found

anonymous · Answer

So I would do $$36(-0.77317)^2-2$$

myininaya · Answer

and what does that give you?
f''(a)>0 implies at x=a there is a min
f''(a)<0 implies at x=a there is a max

myininaya · Answer

is it positive or negative?

anonymous · Answer

That gave me a positive answer, which was 19.52050656

myininaya · Answer

ok so you have f''(a)>0 which implies at x=a there is a min
where our a was -.77317 of course
now that is where the min occurs
now the actual min value

myininaya · Answer

not the actual min value*

myininaya · Answer

f(-.77317) will give you the min value

myininaya · Answer

oh but you don't have to find that

myininaya · Answer

it just asked for the number of max values

anonymous · Answer

Yeah so then how would I find the maximum value with what I've found so far?

myininaya · Answer

or the number of relative extreme

myininaya · Answer

well we just found there was 1

anonymous · Answer

If the only critical point you found turned out to be a min, there is no max.

myininaya · Answer

you remember we determine at -.77317 we had a relative min

anonymous · Answer

So it would be 0?

myininaya · Answer

you know after reading your question again
it seems weird to me

myininaya · Answer

like how far do they want you to go

anonymous · Answer

More like infinity, not 0.

myininaya · Answer

if they wanted you to go as far as finding f'
you could have 3 
if they wanted you go as far as solving f'=0
then it would be 1

anonymous · Answer

Did they ask you to find the relative extreme in a certain range? or of the whole graph?

myininaya · Answer

if they wanted you to go as far as solving f'=0 and verifying it was an actually relative extreme then it would still be 1

anonymous · Answer

Im in an online class through APEX so thats the exact question they asked. It has to be specific answer's. I'm not for sure how far they want me to go, due to not having  a real teacher to tell me, or show me what I'm finding up to. It's all confusing to me.

myininaya · Answer

well there is exactly one relative extrema

anonymous · Answer

This is what I see

myininaya · Answer

The problem I'm having is they wanted you to determine the maximum number of relative extrema that there could be
f'=cubic
cubic has 3 solutions  (so from this step we could say there is at most 3)
f'=0 will only give 1 real solution in this case (so there is at most 1; so it could be 0 or 1 at the step)
f'=0 and then if f''>0 or if f''<0 then we have a relative extrema and this case we did (so we definitely have 1)

anonymous · Answer

I answered with 1, and it told me that was wrong and that the correct answer was 3. Because "This is one less degree of the function."

myininaya · Answer

yeah sorta why we needed to know how far they wanted us to go

myininaya · Answer

f'=cubic tells us there is at most 3 relative extrema

anonymous · Answer

I wasn't for sure myself, because like i said it's an online class with no teacher. I'm never for sure on how far it want's me to go ever.

myininaya · Answer

lol that's kinda irritating i bet

anonymous · Answer

Oh definitely. Especially not learning really how to do what it's"teaching," me. I have to resort to youtube video's 95% of the time!

mathmate · Answer

It $is$ irritating because many of the online questions expect an answer in their current context only.  A proper question should be unambiguous in any context.  :(

myininaya · Answer

So I'm not the only one who found this question unambiguous? That is good to know.

mathmate · Answer

Even though my first reaction was to answer n-1, but then with all the information available, a detailed investigation would not be inappropriate.  So yes I find it ambiguous, the answer could be 0,1 or 3 under different contexts.