How do i prove that a real function is positive?

Answer 1

There are several basic methods. One that works with many elementary functions is to build the function up from simpler functions known to be positive through addition and multiplication. A more general method is to take an arbitrary point x and prove that the function must be positive at that point -- then since x was arbitrary, it follows that the function must be positive at every point. Another method is to assume that there is some point where the function is nonpositive, and derive a contradiction.

A particular case of the last strategy is to prove that a function is continuous, is positive at at least one point x, and is never zero -- then if there were a point y where the function is nonpositive, then (since it is never zero), f(y) is negative, so by the continuity of the function and intermediate value theorem, there is a point c between x and y such that the function is zero there -- but that contradicts the fact that the function is never zero. This strategy is used to establish, for instance, the positivity of the exponential function.

Note that the method suggested by the second poster actually does not work in general -- a function may have a positive integral between any two points, but still fail to be positive. An example is the function f(x) = {1 if x is irrational, -1 if x is rational}. This function is not positive (it is negative at infinitely many points), but for any points a, b with a 0 (note that in this case I am using the Lebesgue integral, since the Riemann integral doesn't exist). Of course, in many practical cases you will be given that the function is continuous, and under that assumption this method does work.

Answer 2

If the integration of that function between any two points is positive then the function is positive.

Answer 3

The prior answer is excellent.

I should add that the most common and elementary functions known to be everywhere positive are exponentials and even powers.

Answer 4

Transformation of linear single-input single-output systems represented by state-variable equations to five canonical forms frequently used in modern control literature is presented in the letter. The necessary and sufficient conditions for a non-singular transformation to exist are brought out in each case.