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Problem 86 Hard Difficulty

For what values of $ c $ does the polynomial $ P(x) = x^4 + cx^3 + x^2 $ have two inflection points? One inflection point? None? Illustrate by graphing $ P $ for several values of $ c $. How does the graph change as $ c $ decreases?

Answer

As the value of $c$ increases, one of the relative minimum points (the one on the left) dips lower and lower while the other (the one on the right) becomes closer and closer to the $x$ -axis.

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Video Transcript

interesting question uh asks about that for what values of cities to polynomial has have to do inflection points or one inflection point and non inflection point. And we have to also illustrate this by graphing P for several values of C. And how the sea changes when how does the graph changes as C decreases? All right. So the inflection point is definitely the point where the double derivative becomes zero and the double derivative changes its sign. So this is the function of the first derivative is going to be p dash X which is four X cube plus three C X square plus two X. And the secondary where there was going to be P double lash X is equal to 12 X square plus six C x plus two. Uh Now this is a quadratic equation of the notice carefully and we need to find uh the values of C. Such that this will have to inflection points of inflection points and nothing but the roots of the equation and to inflection points means we have to roots of this equation. So two rules are possible for a quadratic equation only when they discriminate is positive. One route of the quadratic equation. That is one. Inflection point is possible only when discriminated zero and no inflection point as possible only when they discriminate as negative. So in short we need to find a discriminate and equate uh behavior and find these conditions. So let's find a discriminate first. That's going to be uh six C square minus 4, 12 two. So this is going to be equal to 36 C square minus. Uh This is gonna be four times two is eight and this is 96. Let's see can we take C. Common 60 square minus uh 16 And now can we take to commons? So that's going to be three C square minus eight. So we are left with six times two which is 12 12 times three C square minus eight. So this is a discriminate now for discriminate to be greater than zero. In fact what we can do is we can factories this are discriminated over here only so this is gonna look like let's take three outside as well. So it's gonna look like C square minus 8/3. So it's going to be 36 times C square minus 8/3. So this is going to be 36 times C. Plus route 8/3 C minus route 8/3. Using the difference of the Prophet Square formula. No we're gonna do the discriminative to be in fact we're going to use the sign rule or the the way we go over here. So this is the number line this is how the number line looks like we have negative route 8/3 and route 8/3. So this is the region where this is positive and this is the region where it is negative and this is where these are the points where they discriminate is zero. So these will be the corresponding values of C. So for uh for two inflection points, the range of C is going to be negative infinity to negative negative route 8/3. Union Route 8/3 to infinity For one inflection point for one inflection point to discriminate must be zero. So a C is equal to negative route 8/3 and route 8/3. And for no inflection points, it means that the discriminate is negative so that it has no solution. It means that the value of C is between Negative Route eight or 3 To Route 8/3. So these are the conditions for to have 2, 1 or no inflection point and we also have to graft word. So we'll use this most over here. This is all the graph looks like and uh this is the uh and we are also interested, we are only interested in the double debit of how it changes that sign. So we're gonna graph it, this is how the graph of f double dash X looks like. And clearly we can see that as the value of C approaches approaches. Uh Were to the negative side, it decreases, then the cove moves down and there comes a point when it touches and then intersects the X. Axis. And these students would be the inflection points because this is the these are the roots of the double derivative and as we as see increases it moves up then this is the point where it has no inflection point and this is the range where it has no inflection point, and then over to the right side. Again it intersects it touches and then intersects the X. Axis. Again it has to inflection point. So this is this is matching with our algebraic solution as well. And this is how the cove of uh double derivative changes. And we can also make the same change, saying we can also observe the same change the function. This is how the function is changing. So this is the change in the function in the graph of the function with respect to as the value of C. Changes. Thank you.