Question

1. decide if each set of conditions is possible. if so, give an example. if not, explain why not.

a. a 5th degree polynomial with three turning points
b. a 3rd degree polynomial with three turning points
c. a 4th degree polynomial with three turning points

Explanation:

Step1: Recall the rule for turning - points of a polynomial

The maximum number of turning - points of a polynomial function of degree $n$ is $n - 1$.

Step2: Analyze part a

For a 5th - degree polynomial, $n = 5$. The maximum number of turning - points is $n-1=5 - 1=4$. Since $3\leqslant4$, it is possible. An example is $y=(x - 1)(x + 1)(x - 2)(x+2)(x - 3)$.

Step3: Analyze part b

For a 3rd - degree polynomial, $n = 3$. The maximum number of turning - points is $n - 1=3-1 = 2$. Since $3>2$, it is not possible. A 3rd - degree polynomial can have at most 2 turning - points.

Step4: Analyze part c

For a 4th - degree polynomial, $n = 4$. The maximum number of turning - points is $n - 1=4 - 1=3$. Since $3 = 3$, it is possible. An example is $y=(x + 1)(x - 1)(x+2)(x - 2)$.

Answer:

a. Possible; Example: $y=(x - 1)(x + 1)(x - 2)(x+2)(x - 3)$
b. Not possible; A 3rd - degree polynomial has at most 2 turning - points.
c. Possible; Example: $y=(x + 1)(x - 1)(x+2)(x - 2)$