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haleyelizabeth2017
 one year ago
For which interval(s) is f(x)=x^3+x^25x6 increasing or decreasing?
haleyelizabeth2017
 one year ago
For which interval(s) is f(x)=x^3+x^25x6 increasing or decreasing?

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haleyelizabeth2017
 one year ago
Best ResponseYou've already chosen the best response.0It's not just an "increasing/decreasing" answer lol I have to find the intervals

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1There is something interesting we can actually prove, let's generalize for some point \(a(x_a,y_a)\) on the plane, we have \(f'(a)>0\). This must mean that by definition: \(\lim_{x \rightarrow a} \frac{ f(x)f(x_a) }{ xx_a }\). Then by the theorem of conservation of sign we know that there must exist a \(\delta >0\) that belongs to the interval \(\left xa \right<\delta\) such that \[\frac{ f(x)f(x_a) }{ xx_a }>0\] From here we can conclude two things: (1) if \(x<x_a\) then \(xx_a<0\) which implies that \(f(x)f(x_a)<0\) then \(f(x)<f(x_a)\) (2) if \(x>x_a\) then \(xx_a>0\) which implies that \(f(x)f(x_a)>0\) then \(f(x)>f(x_a)\) by the very definition of increasing or decreasing function we can conclude that function "f" is increasing on the point \(a(x_a,y_a)\). \(if: f'(a)>0 \rightarrow \) f increases on "a". This process is completely analogical to f'(a)<0. Returning to the excercise: \[f(x)=x^3+x^25x6\] What we will do is use the theorem i proved to you earlier, and we will need the derivative of the function in order to apply it: \[f'(x)=3x^2+2x5\] now, we want to know what makes the derivative zero: \[3x^2+2x5=0\]

haleyelizabeth2017
 one year ago
Best ResponseYou've already chosen the best response.05/3 or 1?

freckles
 one year ago
Best ResponseYou've already chosen the best response.1@Owlcoffee I don't think @haleyelizabeth2017 knows any calculus

haleyelizabeth2017
 one year ago
Best ResponseYou've already chosen the best response.0Lol nope...This would be precalc lol

freckles
 one year ago
Best ResponseYou've already chosen the best response.1Calculus is awesome because you can do these questions without a calculator. And it isn't that bad so I guess next year or semester you will get there.

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1Great, now you have to study the sign of that interval. That happens to the function if x takes values greater 1? And what happens if x takes values lesser than 5/3? The study of sign looks like this: dw:1443820022212:dw This means that for the interval \((x; + \infty)\) the function is increasing. for the interval \((\frac{ 5 }{ 3 };1)\) the function is decreasing. for the interval \(( \infty ; \frac{ 5 }{ 3 })\) the function is increasing. you can verify with your calculator.

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1if she's in precalc she must know about derivation, and that's enough to find the variation of a function on the plane @freckles

freckles
 one year ago
Best ResponseYou've already chosen the best response.1I helped her with the previous one. She didn't know derivatives if that is what is meant by the word derivation.

freckles
 one year ago
Best ResponseYou've already chosen the best response.1So that is why I think these are calculator questions instead.

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1I'm sorry, I am from the old school and I did not have those graphing calculators.

freckles
 one year ago
Best ResponseYou've already chosen the best response.1I do like old school stuff. I think calculus is fantastic for these questions and they should probably be saved for calculus class.

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1I personally believe that teachers shouldn't teach with graphing calculators, and start teaching using their brains.

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1but yeah, I already stated the answer... So... Let's move on!

freckles
 one year ago
Best ResponseYou've already chosen the best response.1I wasn't trying to put down your way... I was just saying the op is unfamiliar with that way.

freckles
 one year ago
Best ResponseYou've already chosen the best response.1Your way actually is the exact way I started on her previous question.

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1Don't worry about it, I thought she had the knowledge of derivative. So that's why I started like that.

haleyelizabeth2017
 one year ago
Best ResponseYou've already chosen the best response.0Sadly not :(

Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.1How can someone define increasing or decreasing interval without derivative?... What's wrong with theachers these days?
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