Determine the maximum and minimum number of turning points for the function h(x) = -2x^4 - 8x^3 + 5x -6.
Maximum:3
Minimum:1
Is this a valid reason:
A quartic polynomial function has a 3 Turning points. The turning point is always 1 less than the degree. For example degree 4=3 TP, degree 5=4 turning points?
What about the minimum?
A polynomial of odd degree n has at most n-1 turning points, and may have none.
Think of a line or the curve for y=x^3.
A polynomial of even degree n may have at most n-1 turning points, but must have at least one. Think of a parabola or y=x^4. An even-degree polynomial opens up or down, but must have a min or max.
No, the reason provided for the number of turning points in a quartic polynomial function is not valid. The statement that "the turning point is always 1 less than the degree" is incorrect.
To determine the number of turning points in a polynomial function, we need to consider the behavior of its derivative. A turning point occurs when the derivative changes sign, going from positive to negative or vice versa.
For the function h(x) = -2x^4 - 8x^3 + 5x -6, we need to find the derivative of h(x) and determine the number of sign changes in the derivative.
Taking the derivative of h(x), we get:
h'(x) = -8x^3 - 24x^2 + 5
To find the number of turning points for h(x), we need to find the number of sign changes in h'(x).
Taking a closer look at h'(x), we see that it is a cubic polynomial. Cubic polynomials can have a maximum of 2 turning points.
By observing the graph of h'(x) or by evaluating h'(x) at specific values, we can determine that h'(x) has 1 sign change.
Therefore, the maximum number of turning points of h(x) is 2.
To find the minimum number of turning points, we need to determine if h(x) has any local minimum points. This can be done by analyzing the behavior of h''(x), the second derivative of h(x).
Taking the second derivative of h(x), we get:
h''(x) = -24x^2 - 48x
By examining h''(x), we can see that it is a quadratic polynomial. Quadratic polynomials can have at most 1 turning point.
To determine the number of sign changes in h''(x) and the number of local minimum points, we can evaluate h''(x) at a test point, such as x = 0.
h''(0) = -24(0)^2 - 48(0) = 0
Since the second derivative is equal to 0 at x = 0, we conclude that there is no local minimum point for h(x), and thus the minimum number of turning points is 0.
Therefore, for the function h(x) = -2x^4 - 8x^3 + 5x - 6:
- Maximum number of turning points = 2
- Minimum number of turning points = 0