Local Extrema problem

Question

Find all local extrema of the function f(x)= 1/x-1 + 1/x on the interval (0,1).

Answer 1

Hi Sam,

I understand your problem is f(x) = 1/(x-1) + 1/x is it right? if yes, follow me.

1) x = 1 and x=0 are vertical asymptotes and during interval (0,1) this function is continuous

2) if x approaches to the 0⁺ y goes to the ∞ and when x approaches to the 1^- then y goes to the -∞

3) if x = 1/2 y = 0 it means this function through of x-axis at (1/2,0)

4) on the other hand, this function at given interval is always decreasing because f'(x) = -1/(x-1)² - 1/x² as you see always is negative.

Result: this function has no any critical points and also y will be (-∞, ∞) so there is not any local or absolute maximum or minimum point.

I hope it is useful,

Minoo