How to prove that this function is primitive recursive?

How to prove this function is always negative?

• hello! f(x) = ln (1+x) - x how to prove this function for all x>0 is always negative? i think to use that f is continous and so it has a maximum, but the maximum is 0 and is not ...show more

• Answer:

  Your function is f(x) = ln (1 + x) - x if you differenciate it wrt x, then f'(x) = 1/(1+ x) - 1 now the value of 1/(1+ x) will be always be less than 1 as it is a fractional number for x>0 therefore the derivative of the function is negative for all x>0 and hence the function is decreasing. now, at x = 0, the value of it is ln (1) - 0 = 0 so for every x > 0, the function will have a value less than 0 (as it is decreasing) and it will be negative.