Carkod’s workplace

Mathematics: theorems for functions in economics

Posted on: August 11, 2008

Theorems for functions in economics

Weierstrauss theorem

If f(x) is a continuous function in an interval [a,b], then f(x) achieves a maximum and a minimum in that interval [a,b]. if the interval is unbounded it has no extreme points.

Weierstrauss-bolzano theorem

If f(x) is a continuous function in an interval [a,b], and let f(a)<a, and f(b)>b, or let f(a)>a and f(b)>b, it has point c which belongs to the interval, and valued f(c)=c. That means there´s a function x=y that cuts off f(x).

Bolzano´s theorem (intermediate value theorem).

Let f(x) be a continuous function on a closed interval [a,b], and let k be a number where f(a)<k, f(b)>k, or f(a)>k, f(b)<k, then the function has a point which belongs to the interval, and  f(c)=k.

Bolzano´s theorem of 0’s.

Let f(x) be continuous function in the interval [a,b], let f(a)*f(b)=k , let signs of f(a) and f(b) be different, and if an only if f(a)*f(b)<0, there´s a point c which belongs to the interval [a,b] and makes f(c)=0.

Mean value theorem or Lagrange theorem.

Let f(x) be a continuous function in [a,b], and derivable in [a,b]there´s a point c which belongs to the interval such that f´(x)=f(b)-f(a)/b-a .

Rolle´s theorem.

Let f(x) be a continuous function in [a,b], and derivable in (a,b) and being f(a)=f(b). There´s a point which belongs to (a,b) such that f´(c)=0.

source:my class notes

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