Answer
a. No
b. No
c. 2, 3, 4, 5
d. $y=(x+3)(x+2)(x+1)x(x-1)(x-2)(x-3)$
Work Step by Step
Is it possible for a third-degree polynomial to have exactly one local extremum?
No, the end behavior of this polynomial requires a local maximum and minimum pair or none.
Can a fourth-degree polynomial have exactly two local extrema?
No, the end behavior of this polynomial requires a total of odd number of extrema.
How many local extrema can polynomials of third, fourth, fifth, and sixth degree have?
Answer: 2, 3, 4, 5
Now give an example of a polynomial that has six local extrema.
$y=(x+3)(x+2)(x+1)x(x-1)(x-2)(x-3)$ with 7 zeros and 6 local extrema.
