For the following function f(x)=2x^4-6x^2+2x-1. Find the maximum number of real zeros and Use Descartes's rule of signs to determine how many positive and how many negative zeros the function has. You do not need to find the zeros.
16x^2-1
To determine the maximum number of real zeros of a polynomial function, we can use Descartes's rule of signs.
1. Determine the sign changes in the coefficients of f(x):
- Count the number of sign changes in the polynomial function from positive to negative or from negative to positive. In this case, we have:
- 2x^4: No sign change (positive).
- -6x^2: No sign change (negative).
- 2x: One sign change (positive to negative).
- -1: No sign change (negative).
Therefore, we have one sign change.
2. Determine the possible number of positive zeros:
- The number of positive zeros can either be equal to the number of sign changes or less than that by an even number.
- In this case, we have one sign change in f(x), so the number of positive zeros can be either 1 or 3. It cannot be more than 3 since an even number is subtracted.
3. Determine the possible number of negative zeros:
- Similar to positive zeros, the number of negative zeros can either be equal to the number of sign changes or less than that by an even number.
- In this case, since there is only one sign change, the number of negative zeros can be 1 or 0.
Therefore, based on Descartes's rule of signs, the function f(x) = 2x^4 - 6x^2 + 2x - 1 can have a maximum of:
- 3 positive zeros, and
- 1 negative zero.