if a polynomial function is of degree n, then the graph will have n-1 relative maximum and minimum points.

Is this: Always True, Sometimes True, or Never True?
It's really confusing me.

It is sometimes true. n-1 would be the SUM of the number of relative maxima and minima. It cannot be higher than n-1.

The statement "if a polynomial function is of degree n, then the graph will have n-1 relative maximum and minimum points" is Sometimes True.

This statement is true for polynomial functions of degree n that have distinct real roots. In such cases, the graph of the polynomial function will have n-1 relative maximum and minimum points.

However, this statement is not true for polynomial functions of degree n that have repeated roots or complex roots. In these cases, the number of relative maximum and minimum points may be less than n-1.

Therefore, the statement is sometimes true depending on the nature of the roots of the polynomial function.

To determine if the statement "if a polynomial function is of degree n, then the graph will have n-1 relative maximum and minimum points" is always true, sometimes true, or never true, we need to evaluate its validity.

The statement is sometimes true. Here's why:

For a polynomial function of degree n, the maximum number of relative maximum and minimum points is n-1. However, it is possible for a polynomial function of degree n to have fewer than n-1 relative maximum and minimum points, or even none at all.

Consider a linear polynomial function of degree 1, which is a straight line. It has no relative maximum or minimum points, contradicting the statement.

For a quadratic polynomial function of degree 2, it can have a maximum of one relative maximum or minimum point, which aligns with the statement.

For a cubic polynomial function of degree 3, it can have a maximum of two relative maximum or minimum points, also in accordance with the statement.

However, for higher degree polynomial functions, such as quartic, quintic, or any polynomial of degree greater than 3, the number of relative maximum and minimum points can vary and does not necessarily follow the pattern of n-1.

In summary, while the statement holds true for polynomial functions of degree 2 and 3, it does not hold true for all polynomial functions of higher degrees.