what negative number added to its reciprocal gives the maximum sum

Question

Optimization calculus

pls i need your answers thank you very much

Answer 1

f(x) = x + 1/x  ... function to be maximized, for x < 0.

 

f'(x) = 1 - x^-2; setting this to zero means 1 = 1/x²  and thus x = -1.

 

f"(x) = 2/x³ which is always negative (for negative x) and so this reveals downward concavity of f and thus substantiates that this is a maximum:    f(-1) = 0