Question

2/M and \( m \) denote the local maximum and local minimum values of the faction \( f(x)=x+\frac{1}{x}(x \neq 0) \) respectively, find the value of \( (M-m) \).

Answer 1

f(x) = x + 1/x

for Maxima or Minima  f’(x) = 0

f’(x) = 1 – 1/x²

1 – 1/x² = 0  then x = -1 or 1

f’’(x)  = 2/ x³ 

f’’(1)  = 2 which is  f’’(x)  > 0 , means f(x) will have minima ( local minima ) at x = 1

local minima  m = 1 +1/1 = 2

f’’(-1)  = -2 which is  f’’(x)  <  0 , means f(x) will have maxima ( local maxima ) at x = -1

local maxima   M = -1 +1/(-1) = -2

therefore  M-m = -2 – 2 = -4