Testing Real or Complex Roots: y=g(x)D^{k}f(x)

[Karlisbad]

For the function ## y=f(x) ## is there a test to prove if its roots are real or either has some complex roots?, or in more general cases:

## y=g(x)D^{k}f(x) ## k>0 and a real D=d/dx number.

The question is that sometimes it can be very deceiving to tell if a function has real or complex roots, for example:

## y=exp(2 \pi x)-1 ## has only real roots.. but for real x the function ## y=exp(x^2)+1 ## has only complex roots , but for every real x the function is real.

 

[Mark44]

For the function ## y=f(x) ## is there a test to prove if its roots are real or either has some complex roots?, or in more general cases:

## y=g(x)D^{k}f(x) ## k>0 and a real D=d/dx number.

The question is that sometimes it can be very deceiving to tell if a function has real or complex roots, for example:

## y=exp(2 \pi x)-1 ## has only real roots.. but for real x the function ## y=exp(x^2)+1 ## has only complex roots , but for every real x the function is real.

A simple test is whether the function is always positive or is always negative. In your second example, ##y = \exp(x^2) + 1##, ##x^2 \ge 0## for all real x, and ##\exp(x^2) \ge 1## for all real x, so adding 1 makes ##\exp(x^2) + 1 \ge 2##. Therefore, this function can't have any real roots.