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प्रश्न
For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.
विकल्प
1
`sqrt(3)`
2
None of these
उत्तर
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)`.
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