A polynomial function is a mathematical function that consists of variables and coefficients, using only the operations of addition, subtraction, and multiplication. It can be written in the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is a non-negative integer and an, an-1, ..., a1, a0 are the coefficients.
The degree of a polynomial function is the highest exponent of the variable in the function. For example, in the function f(x) = 3x2 + 5x + 1, the degree is 2.
The maximum or minimum of a polynomial function can be found by finding the critical points of the function, which are the points where the derivative of the function is equal to zero. These points can then be evaluated to determine if they correspond to a maximum or minimum value.
A local maximum or minimum is the highest or lowest point within a specific interval, while a global maximum or minimum is the highest or lowest point over the entire domain of the function.
The first derivative test involves finding the critical points of the function and then evaluating the first derivative at those points. If the first derivative is positive at a critical point, then the point corresponds to a local minimum. If the first derivative is negative, then the point corresponds to a local maximum. If the first derivative is zero, then further analysis is needed to determine if the point corresponds to a maximum or minimum.