write the particular equation of a quartic function that could, according to Descartes Rule of Signs have four positive zeros.
To find a quartic function with four positive zeros, we need to consider Descartes' Rule of Signs. This rule helps us determine the possible number of positive and negative zeros a polynomial function can have.
According to Descartes' Rule of Signs:
1. The number of positive real zeros is equal to the number of sign changes in the coefficients or less by an even integer.
2. The number of negative real zeros is equal to the number of sign changes in the coefficients of f(-x) or less by an even integer.
For a quartic function, the general form is f(x) = ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are coefficients.
Since we want the quartic function to have four positive zeros, there should be zero or two sign changes in the coefficients. This means that the coefficients should have the same sign or alternate signs.
To simplify things, let's consider the case where the coefficients all have the same sign. For example, if all coefficients are positive, we can have:
f(x) = x^4 + x^3 + x^2 + x + c
Where c is a positive constant. This ensures that there are four positive zeros.
However, it's important to note that there are multiple possible equations that satisfy this condition. You can use any positive constant value for c, and it will give you a quartic function with four positive zeros.