Questions: What is the maximum number of relative extrema contained in the graph of this function? f(x) = 3x^5 - x^3 + 4x - 2

Solution Steps

The degree of the polynomial is \(n = 5\).

According to the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has exactly \(n\) roots (real or complex), counting multiplicities.

Descartes' Rule of Signs can give an insight into the number of positive and negative real roots but is not directly related to finding extrema.

The first derivative of the polynomial, \(f'(x)\), determines the critical points. Since we are focusing on the maximum number of relative extrema, we note that a polynomial of degree \(n\) can have at most \(n-1\) turning points.

Since a polynomial of degree \(n\) can have at most \(n-1\) turning points, the maximum number of relative extrema is \(n-1 = 4\).

Final Answer: The maximum number of relative extrema in the graph of the given polynomial function is 4.