Question

8 list the possible number of extrema for each type of polynomial. a 3rd - degree polynomial b 4th - degree polynomial

Explanation:

Step1: Recall the formula for the number of extrema

The number of extrema of a polynomial function \(y = f(x)\) is at most \(n - 1\), where \(n\) is the degree of the polynomial.

Step2: For a 3rd - degree polynomial

Let \(n=3\). Using the formula \(n - 1\), we get \(3-1 = 2\). A 3rd - degree polynomial can have \(0\) or \(2\) extrema. For example, \(y=x^{3}\) has \(0\) extrema (\(y^\prime=3x^{2}\geq0\), critical point at \(x = 0\) which is not a local maximum or minimum), and \(y=x^{3}-3x\) has \(y^\prime=3x^{2}-3=3(x + 1)(x - 1)\), with two extrema (a local maximum at \(x=-1\) and a local minimum at \(x = 1\)).

Step3: For a 4th - degree polynomial

Let \(n = 4\). Using the formula \(n-1\), we get \(4 - 1=3\). A 4th - degree polynomial can have \(1\) or \(3\) extrema. For example, \(y=x^{4}\) has \(y^\prime=4x^{3}\), critical point at \(x = 0\) which is a local minimum (one extremum). And \(y=x^{4}-2x^{2}\) has \(y^\prime=4x^{3}-4x=4x(x^{2}-1)=4x(x + 1)(x - 1)\), with three extrema (a local maximum at \(x = 0\) and local minima at \(x=\pm1\)).

Answer:

a. A 3rd - degree polynomial can have \(0\) or \(2\) extrema.
b. A 4th - degree polynomial can have \(1\) or \(3\) extrema.