Question

At most, how many times could the graph of f(x) = -ax⁶ + bx⁵ + cx⁴ + dx³ - ex² - fx - g change direction?

Answer 1

The graph of
`\( f(x) = -ax^6 + bx^5 + cx^4 + dx^3 - ex^2 - fx - g \)` could change direction at most
`\( 5 \)` times.

Step-by-step explanation:

The number of times a polynomial function can change direction is determined by the number of its real roots, considering multiplicities. In this case, the given function is a sixth-degree polynomial, meaning it can have at most
`6` real roots. The graph changes direction at each real root, and the maximum number of changes in direction is 5 because the leading term
`\(-ax^6\)` results in an even degree, and the sign of the function does not change at the highest point of each hump or the lowest point of each dip.

To comprehend this, consider the behavior of the function as
`\( x \)`approaches positive or negative infinity. The leading term
`\( -ax^6 \)`dominates, and the function tends to
`\(-\infty\) as \( x \)` approaches either positive or negative infinity. This means that there is an even number of turning points. Since the function has
`\(6\)` roots, the maximum number of times it can change direction is
`\(5\)`.

In summary, the even degree of the leading term in the polynomial restricts the number of times the graph changes direction, and in this case, it is
`\(5\)` times, corresponding to the
`\(5\)` possible real roots of the sixth-degree polynomial function.