Use the intermediate value theorem to show that each polynomial f...

Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=3x^2-x-4; 1 and 2

Accepted Answer

Hey, everyone in this problem, we're asked to express that the given function has a real zero between the X values 12 and 14 using the intermediate value zero. The function we're given is F of X is equal to X squared minus 19 X plus 78. We're given four answer choices, options A through D that give the function value of the two end points 12 and 14 and indicate whether that's positive or negative. And we're gonna come back to those as we work through this problem. So the first thing we wanna do in order to use the intermediate value theorem, like the question is asking is to figure out whether we can actually apply it and are the conditions of the theorem satisfied. Now, for the intermediate value theorem, what we need to look at is whether it is continuous on the closed interval that we're given. OK. So here we have a polynomial. Our function F FX is a polynomial. We know it is continuous everywhere. And so we can say that F FX is continuous on the closed interval from 12 to 14. OK. Those X values we're interested in. So the intermediate value the mole short form is I V T applies. All right. So now we at least know that we can apply this theorem. Let's think about what it's gonna tell us. OK. If the intermediate value theorem applies, it means that there exists right. A value C such that if we take the minimum function value between our end points F an F-14 and we take the maximum function value between our end points F 12 and F-14. We will have some value in between called F F C. OK? So there's some X value that will give us a function value that's between the minimum and the maximum. All right. So if we want to figure out what that minimum and maximum are, we need to find the function evaluated at the end points. OK? Now we got a hint of that when we looked at our answer choices that we need to, to evaluate our function at these points. So starting with the left end point, we have F at 12. OK? We're gonna substitute X is equal to 12 everywhere. In this function, we have 12 squared minus multiplied by 12 plus 78. This is equal to 144 minus 228 who was which gives us a function value F of 12, which is equal to negative six. OK? And we're gonna note that this is less than zero. This is a negative value. Now, looking at the other end point, we have the F of 14 is equal to 14 squared minus 19, multiplied 19, sorry, not 12. Ok. So 19 multiplied by 14 plus 78. Again, simplifying this 196 minus 266 plus 78 which gives us a function value F F- that is equal to eight. And we're gonna note that this is positive, this is greater than zero. So what we have here is we have one value that's negative and one value that's posi and we want to know whether this guarantees zero. And in this case, it will, OK? If we have one value negative, one value positive and the function is continuous between those two points, it must pass through zero at some point. OK? And so we get that we have a guaranteed zero between these two points. And let's compare this to our answer choices. OK? We found that the value F of 12 was equal to negative six. We can eliminate option C and D because they both had positive six. We found that the function F evaluated at 14 was equal to positive A. OK. So this corresponds with answer choice A because we had one negative value and one positive value. The intermediate value theorem guarantees a zero between those two points. Thanks everyone for watching. I hope this video helped see you in the next one.

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Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=3x^2-x-4; 1 and 2