For what values of c does the polynomial P(x) = x^4 + cx^3 + x^2 have two inflection points? One

step 1 Interesting question. It asks about that for what values of C does this polynomial have two iden

For what values of c does the polynomial P(x) = x^4 + cx^3 +...

Key Concepts

Discriminant of a Quadratic Equation

When the second derivative is a quadratic expression in x, the discriminant (the part under the square root in the quadratic formula) determines the nature of its roots. A positive discriminant implies two distinct real roots, indicating two potential inflection points, a zero discriminant implies a repeated real root (which may be an inflection point if confirmed by a sign change), and a negative discriminant implies no real roots, hence no inflection points. This tool is essential in analyzing how the parameter in the polynomial affects the number and nature of inflection points.

Parameter Variation and Graph Transformation

The parameter in a polynomial function, such as 'c' in the given function, can drastically alter the shape of the graph by affecting the location and number of inflection points as well as the overall curvature. Understanding how variations in this parameter influence the function's second derivative allows one to predict how the graph’s concavity changes, which in turn informs about shifts in the graph's turning behavior and symmetry.

Inflection Points

Inflection points are the points on a graph where the concavity changes. In the context of polynomial functions, an inflection point occurs where the second derivative either crosses zero and changes sign or is undefined and changes sign. This concept is crucial for understanding the curvature and shape of the graph of the function as parameters change.

Second Derivative Analysis

The second derivative of a function provides information about its concavity. For a polynomial function, setting the second derivative equal to zero helps locate potential inflection points. The analysis of the second derivative not only involves solving for its zeros but also checking whether the sign of the second derivative actually changes at those points, which confirms the presence of an inflection point.

Transcript

0:00 Interesting question.

00:01 It asks about that for what values of c does this polynomial have two identixtion points or one inflection point and non -infliction point.

00:11 And we have to also illustrate this by graphing p for several values of c and how does c changes when, how does the graph changes as c decreases? all right.

00:23 So the infliction point is definitely the point where the double derivative becomes a zero and the double derivative changes its sign so this is the function the first derivative is going to be p dash x which is 4x cube plus 3 cx square plus 2x and the second derivative is going to be p double dash x is equal to 12x square plus 6 cx plus 2 now this is a quadratic equation if we notice carefully and we need to find the values of c such that this will have two inflection points so inflection points are nothing but the roots of the equation and two inflection points means we have two roots of this equation so two roots are possible for a quadratic equation only when the discriminant is positive one root of the quadratic equation that is one inflection point is possible only when discriminant is zero and no inflection point is possible only when the discriminant is negative.

01:29 So in short, we need to find a discriminant and equate it and behave and find these conditions.

01:35 So let's find a discriminant first.

01:37 That's going to be 6c square minus 4 .12.

01:43 So this is going to be equal to 36 c square minus this is going to be 4 times 2 is 8 and this is 96.

01:53 Let's see.

01:54 Can we take c common? 6c square minus 16 and now can we take two common so that's going to be 3c square minus 8 so we are left with 6 times 2 which is 12 12 times 3 c square minus 8 so this is the discriminant now for discriminant to be greater than 0 in fact what we can do is we can factorize this a discriminant over here only so this is going to look like let's take three outside as well so it's going to look like c square minus 8 over 3 so this is going to be 36 times c square minus 8 over 3 so this is going to be 36 times c plus root 8 over 3 c minus root 8 over 3 using the difference of the perfect square formula now we are going to do the discriminant to be in fact we are going to use the sign rule or the the wavy curve over here so this is the number line this is how the number line looks like we have negative root 8 over 3 and root 8 over 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 the discriminant is zero so these will be the corresponding values of c so for for two inflection points the range of c is going to be negative infinity to negative to negative negative root 8 over 3 union root 8 over 3 to infinity for one inflection point for one inflection point a discriminant must be 0 so c is equal to negative root 8 over 3 and root 8 over 3 and for no inflection points it means that the discriminant is negative so that it has no solution it means that the value of c is is between negative root 8 over 3 to root 8.

04:05 So these are the conditions for it to have 2 -1 on no inflection point.

04:12 And we also have to graph it.

04:13 So we'll use dismos over here...

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