Advertisements
Advertisements
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
Solution
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)`.
APPEARS IN
RELATED QUESTIONS
Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]
Verify Rolle's theorem for the function
f(x)=x2-5x+9 on [1,4]
Check whether the conditions of Rolle’s theorem are satisfied by the function
f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3]
Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?
f (x) = [x] for x ∈ [5, 9]
Verify Mean Value Theorem, if f (x) = x2 – 4x – 3 in the interval [a, b], where a = 1 and b = 4.
Verify Mean Value Theorem, if f (x) = x3 – 5x2 – 3x in the interval [a, b], where a = 1 and b = 3. Find all c ∈ (1, 3) for which f ′(c) = 0.
Verify Rolle’s theorem for the following function:
f (x) = x2 - 4x + 10 on [0, 4]
Verify Rolle’s theorem for the following function:
`f(x) = e^(-x) sinx " on" [0, pi]`
f(x) = (x-1)(x-2)(x-3) , x ε[0,4], find if 'c' LMVT can be applied
Verify Langrange’s mean value theorem for the function:
f(x) = x (1 – log x) and find the value of c in the interval [1, 2].
Verify Rolle’s Theorem for the function f(x) = ex (sin x – cos x) on `[ (π)/(4), (5π)/(4)]`.
Verify Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]`.
The value of c in Rolle’s Theorem for the function f(x) = e x sinx, x ∈ π [0, π] is ______.
f(x) = `x(x + 3)e^((–x)/2)` in [–3, 0]
Discuss the applicability of Rolle’s theorem on the function given by f(x) = `{{:(x^2 + 1",", "if" 0 ≤ x ≤ 1),(3 - x",", "if" 1 ≤ x ≤ 2):}`
Find the points on the curve y = (cosx – 1) in [0, 2π], where the tangent is parallel to x-axis
Using Rolle’s theorem, find the point on the curve y = x(x – 4), x ∈ [0, 4], where the tangent is parallel to x-axis
f(x) = sinx – sin2x in [0, π]
f(x) = `sqrt(25 - x^2)` in [1, 5]
Find a point on the curve y = (x – 3)2, where the tangent is parallel to the chord joining the points (3, 0) and (4, 1)
Using mean value theorem, prove that there is a point on the curve y = 2x2 – 5x + 3 between the points A(1, 0) and B(2, 1), where tangent is parallel to the chord AB. Also, find that point
If `1/(a + ω) + 1/(b + ω) + 1/(c + ω) + 1/(d + ω) = 1/ω`, where a, b, c, d ∈ R and ω is a cube root of unity then `sum 3/(a^2 - a + 1)` is equal to
P(x) be a polynomial satisfying P(x) – 2P'(x) = 3x3 – 27x2 + 38x + 1.
If function
f(x) = `{{:((P^n(x) + 18)/6, x ≠ π/2),(sin^-1(ab) + cos^-1(a + b - 3ab), x = π/2):}`
is continuous at x = ` π/2`, then (a + b) is equal to ______.
