Final answer:
The function f(x) = -ax⁶ + bx⁵ - cx⁴ + dx³ + ex² - fx + g can change direction at most 4 times.
Step-by-step explanation:
The function f(x) = -ax⁶ + bx⁵ - cx⁴ + dx³ + ex² - fx + g can change direction at most 4 times.
To determine the number of times the graph can change direction, we need to analyze the behavior of the polynomial function. Each change in direction corresponds to a change in concavity. The highest power of x in the function is 6, so the highest degree of the polynomial is 6. Since a polynomial of degree n can have at most n-1 changes in concavity, the function can change direction at most 6-1 = 5 times. However, this assumes that all the coefficients (a, b, c, d, e, f, g) are nonzero and have different signs.
If some of the coefficients are zero or have the same sign, the number of changes in direction decreases. Therefore, the function f(x) = -ax⁶ + bx⁵ - cx⁴ + dx³ + ex² - fx + g can change direction at most 4 times.