For the function f(x) = x + 1/x A. x = 1 is a point of maximum B. x = –1 is a point of minimum ​

Question

For the function f(x) = x + \(\frac{1}{x}\) 

A. x = 1 is a point of maximum 

B. x = –1 is a point of minimum 

C. maximum value > minimum value 

D. maximum value < minimum value

Answer 1

Option : (C)

In such type of questions 

find both the maximum and minimum value to compare the options.

So first,

f(x) = x + \(\frac{1}{x}\)

So,

f'(x) = 1 - \(\frac{1}{x^2}\) 

put f’(x) = 0,

1 - \(\frac{1}{x^2}\) = 0

1 = \(\frac{1}{x^2}\) 

⇒ x = ±1 

Hence by second derivative test 

f’’(x) > 0 or f”(x) < 0 

so it’s a point of minimum or maximum respectively.

f"(x) = \(\frac{2}{x^3}\),

f”(-1) = -20 

So x = 1 is a point of minimum and 

x = -1 is a point of maximum 

f(1) = 2 is minimum value. 

f(-1) = -2 is maximum value. 

Therefore, 

maximum value < minimum value.