Prove the equation has at least one real root (KristaKingMath)

Learn how to use the Intermediate Value Theorem to prove that the function as at least one real root. To do this, you'll need to look at the range of each term in the function, and then use that information to pinpoint values you should focus on. You'll need to look for points above and below the x-axis. If you can prove that a point exist below the x-axis, and that another point exists above the x-axis, and you know that the function is continuous between those points, then you can use the Intermediate Value Theorem to prove that the function must assume a value of zero somewhere between those two points. In other words, the function must have a real root between these points.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[omkars]

After you say "Hi everyone" it seems as if you wait for us to say hi back, lol.  Thanks for all your videos.

[cardiacmyxoma]

literally youre saving me right now, i have a midterm tomorrow morning and im panicking! thank you so so so so much and i really do hope you continue to make vids. xx

[kristakingmath]

i'd recommend finding the domain on which it IS continuous, and seeing if you can find a root there. if not, check the other continuous sections. :)

[pachinkoly]

THANK YOU SO MUCH FOR YOUR VIDEOS!! I wish I discovered them earlier in the year. I will be taking the calc exam in two days and I was really worried about it, but your videos have helped me feel more confident, because they're so easy to understand and concise. Thank you again, you're a lifesaver :)

[bestyoueverhad.]

For the first time i understand this. Thank you very much! thumbs up.

[ThePinoyMamba]

Thanks. This is hard to prove sometimes. But now is clearer. Because just saying like words. Now its way better

[Zeitaluq]

Quick query. When you use the f(x) notation and substitute x = 2, x = 1.1 and x = -2; we are keeping Cosine (x) as 1 or -1 on respective maximum and minimum values?

Being literal means subtitution x = 1.1 as (1.1)^3 - cosine (1.1) although I guess the extreme high (or respective low) values of the function are being used to prove the Intermediate?

[Borz-RAI]

A harder example would be to prove that a function has at most one root in a given interval. In that case proof by contradiction would work best.

[TheDiederikdehaan]

I used the interval [0, pi] because I know that the cosine fluctuates between 1 and -1 in that domain. Is that also a way to prove the above mentioned statement: f(0)<0<f(pi) --> -1<f(c)< pi^3-pi?

[Unnjit]

Wow, I come here because I'm stuck on a question. This is the exact question. You're very helpful.

[mwesigwapeter]

Best thing i have come across this year so far, all the lessons are so helpful. lots of thanks

[ninoz]

but how do we deal with equations like 100e^(-x/100) = 0.01x^2 ??

[karrobat]

What does it matter if the range is 1 to -1 if we're talking about x inputs? Doesn't range refer to y? I'm confused :(

[JD-cyqh]

For the last part it shpuld be f(any number less than -1)<0<f(any number more than 1). Right??

[omidheidari]

hey, you are mixing the range and the domain of Cos(x).

[TheDramaticNerd]

I realized that you're technically saying that cos (1) is 1 and cos (-1) is -1, because you're using f(1) and f(-1). So, instead of the root being in the interval f(-1) < 0 < f(1), shouldn't it be in f(0) < 0 < f(pi)? Or -1 < 0 < 1?

[evildoctorbluetooth]

This reminds me of my first year of uni. I had forgotted most of it. Thank you for reminding me of these mathematical things.

[khadijahflowers]

Does it matter which side you put the terms on? For example, instead of of y = x^3 - cosx could you get the same answer if you had subtracted x^3 to get y = cosx - x^3 ? It looks like you would get the opposite signs on opposite sides. Would that still count when you went to prove it?

[YeshuaLives]

she explains it waaay better than my prof! thank you muuuch!

[Abhisruta]

You could use graph to solve this question easier...take cos x = x^3 = y and count the common pts., which are roots.