Is it possible for a third-degree polynomial to have exactly...

Name: Problem 92, Chapter 3: College Algebra
Uploaded: 2020-07-31T06:30:31-08:00
Duration: 4 min 28 s
Channel: Ankit Gupta
Description: Is it possible for a third-degree polynomial to have exactly one local extremum? Can a fourth-degree polynomial have exactly two local extrema? How many local extrema can polynomials of third, fourth, fifth, and sixth degree have? (Think about the end behavior of such polynomials.) Now give an example of a polynomial that has six local extrema.

Is it possible for a third-degree polynomial to have exactly one local extremum? Can a fourth-degree polynomial have exactly two local extrema? How many local extrema can polynomials of third, fourth, fifth, and sixth degree have? (Think about the end behavior of such polynomials.) Now give an example of a polynomial that has six local extrema.

Key Concepts

End Behavior of Polynomials

The end behavior of a polynomial, determined by the polynomial’s degree and the sign of its leading coefficient, describes how the function behaves as x approaches positive or negative infinity. This behavior constrains the possible arrangement of local extrema on the graph, since the function must eventually conform to its dominant term, affecting the number and placement of turning points.

Constructing Polynomials with Given Extrema

Constructing a polynomial with a specified number of local extrema involves ensuring that the derivative of the polynomial has the appropriate number of real, distinct roots, with each corresponding to a genuine extremum confirmed by a change in the sign of the derivative. This process may involve designing polynomials with factors that are arranged to produce the maximum number of turning points (or fewer if desired), in adherence with the restrictions imposed by the degree of the polynomial.

Polynomial Degree and Turning Points

The degree of a polynomial sets an upper bound on the number of turning points it may have; specifically, a polynomial of degree n can have at most n-1 turning points. This is a central concept as it establishes the maximum possible number of local extrema (maximums or minimums) that a polynomial function can exhibit, though the actual number may be lower due to the nature of the critical points.

Local Extrema and Critical Points

Local extrema are the points on a graph where the function reaches a local maximum or minimum. They occur at critical points—points where the first derivative is zero or undefined—and a change in the sign of the derivative confirms an extremum. Understanding the relationship between a function’s critical points and its local extrema is essential to analyzing and predicting the behavior of polynomial graphs.

Transcript

00:01 So, in this question we have a few number of parts, so i will answer each of them serial wise.

00:10 So is it possible for a third degree polynomial to have exactly one local extrema? no, it's not possible.

00:18 It's not possible for a three degree polynomial or i can say cubic to have exactly one local extrema.

00:38 Either it will have no local extremer or it will have two local extremes.

00:42 Next in the second part, can a 4 degree polynomial have exactly two local extremar? again, it's a no.

00:50 4 degree polynomial cannot have exactly two local extremal.

01:18 In the next part, how many local extrema can a polynomial have? 3 degree, 4, 3 degree, 3 degree and 6 degree have? so i will write number of local extrema.

01:32 Extrema a polynomial of first one three degree so for a three degree it can have two local extrema for a four degree polynomial it can have three or one local extrema for a five degree polynomial it can have either four or one local extrema for a five degree 4 local extremar and for a 6 degree polynomial there can be 5, 3 or 1 local extremer.

02:41 Now in the next part they are asking a polynomial that has 6 local extrema.

02:47 So, i will say for a polynomial with 6 local extrema you need to consider 7 degree polynomial.

03:18 So at least 7 degree polynomial must be there.

03:21 So just for an example i can consider anything like this x minus a x minus b h minus c x minus b x minus e this is 5 x minus f and x minus g if i call this as p x this is a 7 degree polymer will be having all of its roots...

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