Question

x and y be two variables such that x > 0 and xy = 1, then the minimum value of (x + y) is
1. 2
2. 3
3. 4
4. 0

Answer 1

Correct Answer - Option 1 : 2

Concept:

Minima and maxima of a function: These are the value of functions where its first derivative (slope) becomes zero.

We can calculate the minima or maxima of function f(x) by the following steps.

Step:1 Calculate the first derivative f'(x) of a function.

Step:2 Equate it with zero and find the critical values (let x1 and x2).

Step:3 Find out the second derivative f''(x) and check its sign on critical values.

Step:4 If f''(x) < 0 then there will be maxima and if f''(x) > 0 than minima at corresponding value.

Formula used:

\(\frac{d}{dx}x^n\ =\ nx^{n-1}\)

Calculation:

Let, f(x) = x + y

According to question, 

xy = 1 ⇒ y = 1/x

Therefore,

f(x) = x + 1/x   ----(1)

Diff. with respect to x

f'(x) = 1 - 1/x²

f''(x) = 2/x³

For maxima and minima,

1 - 1/x2 = 0

⇒ x = ± 1

But According to question, x > 0

⇒ x = 1

Hence, from equation (1), minimum value of f(x) is

= 1 + 1/1

⇒ (x + y)_min = 2