College Algebra, Chapter 4, 4.2, Section 4.2, Problem 80

a.) How many $x$-intercepts and local extrema does the polynomial $P(x) = x^3 - 4x$ have?
b.) How many $x$-intercepts and local extrema does the polynomial $Q(x) = x^3 + 4x$ have?
c.) Suppose that $a > 0$, how many $x$-intercepts and local extrema does each polynomials $P(x) = x^3 - ax$ and $Q(x) = x^3 + ax$ have? Explain your answer.

a.)


Based from the graph $P(x) = x^3 - 4x$ has 3 $x$-intercepts and 2 local extrema.
b.)


Based from the graph $Q(x) = x^3 + 4x$ has 1 $x$-intercept and 0 local extrema.
c.) The number of $x$-intercepts and local extrema always depends on the number degree of the function. The maximum number of $x$-intercepts is equal to the degree of the function. While the number of the local extrema can never be greater than $n -1$, where $n$ is the degree of the function. This is true for all functions. However, it depends on the orientation of the function. For example, like $Q(x) = x^3 + 4x$, the function is always increasing, that's why it has only 1 $x$-intercept and 0 local extrema but the number of the $x$-intercept does not exceed the degree of the function as well as the local extrema.