Question

Examine the function f(x) = `x + 25/x ` for maxima and minima 

Answer 1

f(x) = `x + 25/x`

`f'(x) = 1 - 25/x^2`

`f''(x) = 50/x^3`

f has maxima or minima if 

f'(x) = 0

`1 - 25/x^2 = 0`

`x^2 - 25 = 0`

∴ x = 5 , x = -5

`f''(5) = 50/5^3`

`f''(5) = 2/5 > 0`

`f''(-5) = 50/(-5)^3`

`f''(-5) = (-2)/5 < 0`

∴ From equation (ii) f has maxima at x = - 5. And maximum value of f is 

`f_(max) = f(-5)`

= `-5 + 25/-5`

= -10

Also from equation (i) f has minima at x = 5 
And minimum value of f is 

`f_(min) = f(5)`

 = `5 + 25/5`

`f_(min) = 10`