What is the least amount of extrema (relative min/max) a 6th degree polynomial can have?
one
consider y = x^6
To determine the least amount of extrema a 6th-degree polynomial can have, we need to consider its degree and the behavior of the polynomial function.
A 6th-degree polynomial has the form:
f(x) = ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g
The number of extrema is related to the number of changes in concavity. A change in concavity occurs at a point where the second derivative changes sign.
Let's analyze the polynomial's behavior:
1. The degree of the polynomial is 6, which means it can have up to 6 zeros or roots.
2. For a polynomial of degree n, the number of extrema can be at most (n - 1). So, in this case, the maximum number of extrema is (6 - 1) = 5.
3. To determine the least number of extrema, we need to consider whether the polynomial is "smooth" enough to have extrema. For example, a polynomial with all real roots will not have any extrema.
In summary, the least amount of extrema a 6th degree polynomial can have is 0 if it has all real roots. However, if the polynomial has complex roots or multiple roots, it is possible to have up to 5 extrema.