For the defined function f(x) on the interval I, if x0 belongs to I, so that any x belongs to I, there is the following inequality:
f(x)<=f(x0), or
f(x )>=f(x0),
then call f(x0), which is the maximum or minimum value of function f(x) in interval I.
The maximum and minimum values can be equal.
Theorem 1: Boundedness and Maximum and Minimum Theorem
A function that is continuous on a closed interval && is bounded on the interval, and the maximum and minimum values must be obtained.
Keywords: ratio interval, continuous, bounded.
Theorem 2: Zero Theorem
If x0 makes f(x0)=0, then x0 is called the zero point of function f(x).
Suppose that the function f(x) is continuous on the closed interval [a,b], and f(a) and f(b) have different signs, then there is at least some ξ Cauchy in the open interval (a,b), so that f(ξ) )=0
Theorem 3: Boundary Value Theorem
Let the function f(x) be continuous on the closed interval [a,b], and take different function values at the end points of this interval
f(a)=A
f(b)=B,
then for any one between A and B Number C, there is at least a little ξ in the open interval (a, b), so that:
f(ξ)=C
a<ξ<b