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?
Haha, I appreciate your creativity, but let me clown around with this question a bit. The explanation you provided for the maximum turning points is kind of humorous, but unfortunately not quite accurate. The number of turning points in a quartic function, like h(x) = -2x^4 - 8x^3 + 5x -6, can actually vary.
To determine the maximum number of turning points, you can look at the degree of the function. Since h(x) is a quartic function (degree 4), the maximum number of turning points is indeed 3.
Now, let's talk about the minimum turning points. Brace yourself for a tiny anticlimax, because regardless of the degree of the polynomial, the minimum number of turning points for any polynomial function is always 0. Yeah, I know, it's a bit of a letdown. But hey, at least we have our maximum turning points to make up for it!
So, to sum it up: the maximum number of turning points for h(x) = -2x^4 - 8x^3 + 5x -6 is 3, and the minimum number of turning points is 0. Keep in mind that these are just the maximum and minimum possibilities, and the actual number of turning points may vary.
No, the reason provided is not valid for determining the number of turning points. The statement "The turning point is always 1 less than the degree" is incorrect. The number of turning points of a polynomial function is not directly determined by the degree of the polynomial.
To determine the maximum and minimum number of turning points for the given function h(x) = -2x^4 - 8x^3 + 5x - 6, we need to consider the behavior of the function and its derivatives.
First, let's find the derivative of h(x):
h'(x) = -8x^3 - 24x^2 + 5
To find the number of turning points, we need to find the number of real roots of the derivative h'(x).
By using calculus or graphical analysis, we can determine that the derivative h'(x) has 2 real roots. Since h(x) is a quartic polynomial, it can have at most one additional turning point, which is either a local maximum or minimum.
Therefore, the maximum number of turning points for h(x) is 3 (2 real roots + 1 additional turning point), and the minimum number of turning points is 1 (2 real roots - 1 additional turning point).
The reason provided for determining the maximum number of turning points is not entirely accurate. The number of turning points of a polynomial function is equal to the degree minus one only if the polynomial function is a power function (i.e., of the form f(x) = ax^n, where a is a constant and n is a positive integer). However, the given function, h(x) = -2x^4 - 8x^3 + 5x - 6, is not a power function since it is a quartic polynomial with additional terms involving x^3, x, and a constant.
To determine the maximum number of turning points, we can look at the degree of the polynomial function. The degree of a polynomial is the highest power of the variable in the function. In this case, the highest power of x in h(x) is x^4, which means the degree of the polynomial is 4.
For a quartic polynomial function (degree 4), the maximum number of turning points is four. However, this does not mean that every quartic polynomial will have four turning points. It only means that there can be at most four turning points. In the case of h(x) = -2x^4 - 8x^3 + 5x - 6, it may or may not have four turning points.
To determine the actual number of turning points, we would need to analyze the behavior of the function in more detail, such as finding the critical points and performing a second derivative test. However, the given question does not provide enough information or context to determine the exact number of turning points.
Regarding the minimum, the given function h(x) = -2x^4 - 8x^3 + 5x - 6 does not necessarily have a local minimum. To find the local minimum of a function, we typically need to find the critical points and perform a second derivative test to check if they correspond to a minimum. However, without further information or context, we cannot determine the minimum value or whether it exists for the given function.
If the x^4 term is positive, then the curve rises in the 1st and 2nd quadrants, so there has to be at least one minimum point.
if the x4 term is negative, then the curve drops down in the 3rd and 4th quadrants, and there has to be at least one maximum.
For yours the curve will be downwards, so it could look like an upside down W
Draw a WW. You will see that there are 2 mins and 1 max if it opens up, and if the curve looks like an M, (-x^4), there could be 2 max's and 1 min
There cannot be 3 max's and 1 min for a quartic
Your statement about the turning points is correct, but those turning points could be maximums or minimums
I will give you a few examples of equations where it is obvious, I will leave the equations in factored form
1. y = (x+1)(x+2)(x-3)(x+4)
The curve opens up , and has x-intercepts at
x = -1,-2,3, and 4 , so it is easy to see that there must be 2 mins and 1 max
2. y = (x-2)(x-3)(x^2 + 4)
http://www.wolframalpha.com/input/?i=+%28x-2%29%28x-3%29%28x%5E2+%2B+4%29%3D0
3. y = (x-10)(x^3-3)
http://www.wolframalpha.com/input/?i=+%28x-10%29%28x%5E3-3%29%3D0
Play around with different quartics on this webpage, and you can see the different cases.
try:
y = x^4
y = x^4 + x^3
y = x^4 + x^3 + x^2
etc
now change some of the signs .
Have some fun with math.