In Exercises 33–40, use the Intermediate Value Theorem to show th...

Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^3−4x^2+2; between 0 and 1

Accepted Answer

Welcome back. I'm so glad you're here. We are given this polynomial function F of X equals to X. Cute minus eight X squared plus four. And we're told that it has a real zero in between negative one and zero. And we are to prove this using the intermediate value theorem. Well, we recall from previous lessons that the intermediate value theorem tells us that if F of negative one and F of zero have opposite signs, then at least one real zero lies in between negative one and zero. So let's test this by figuring out the sign of F of negative one. We're going to plug in negative one in for X everywhere there was an X in our original F of X function. So this will be two times negative one cubed minus eight times negative one squared plus four. Let's work on the exponents bring down that two times and negative one cubed is negative, one minus eight times negative one squared is going to be a positive one plus four. Now let's do the multiplication two times negative one is negative two minus eight times one is going to be a minus eight plus four. Doing that. Addition and subtraction negative two minus eight, negative 10 plus four gives us a negative six. So F of negative one equals negative six. And note that that is a negative value. And now for the intermediate value theorem we now get to test out F of zero and see if that will give us a sign change. We're going to plug zero in everywhere. We had an X. In our original F of X equation. So now it's two times zero. Cute minus eight times zero squared plus four. While these first two terms that's just going to be zero. So our F of zero equals four. And note that yes, this is a positive solution. So we did in fact have a sign change. And the intermediate value theorem tells us that yes, there is at least one real zero in between negative one and zero because F of negative one and F of zero gave us a sign change. We look at our answer choices in this matches with answer choice B. Well done. We'll catch you on the next one.

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In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^3−4x^2+2; between 0 and 1