Question

question 4 of 25
what is the maximum number of relative extrema contained in the graph of
this function?

f(x) = 3x⁴ - x² + 4x - 2

Explanation:

Step1: Recall the rule for relative extrema

For a polynomial function \( f(x) \) of degree \( n \), the maximum number of relative extrema is \( n - 1 \). This is because the derivative \( f'(x) \) is a polynomial of degree \( n - 1 \), and the number of critical points (where \( f'(x)=0 \) or is undefined) is at most \( n - 1 \), and relative extrema occur at critical points.

Step2: Determine the degree of the given function

The function \( f(x)=3x^{4}-x^{2}+4x - 2 \) is a polynomial function. The highest power of \( x \) is 4, so the degree \( n = 4 \).

Step3: Calculate the maximum number of relative extrema

Using the rule from Step 1, the maximum number of relative extrema is \( n-1=4 - 1=3 \).

Answer:

3