Question

For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.

Options

• 1

• `sqrt(3)`

• 2

• None of these

Answer 1

For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is `sqrt(3)`.

Explanation:

Given that: f(x) = `x + 1/x`, x ∈ [1, 3]

We know that if f(x) =  `x + 1/x`, x ∈ [1, 3] satisfies all the conditions of mean value theorem then

f'(c) = `("f"("b") - "f"("a"))/("b" - "a")` where a = 1 and b = 3

⇒ `1 - 1/"c"^2 = ((3 + 1/3) - (1 + 1/1))/(3 - 1)`

⇒ `1 - 1/"c"^2 = (10/3 - 2)/2`

⇒ `1 - 1/"c"^2 = 4/6 = 2/3`

⇒ `- 1/"c"^2 = 2/3 - 1`

⇒ `- 1/"c"^2 = -1/3`

⇒ `1/"c"^2 = 1/3`

⇒ c = `+- sqrt(3)`.

Here c = `sqrt(3) ∈ (1, 3)`.