Advertisements
Advertisements
प्रश्न
Verify the Lagrange’s mean value theorem for the function:
`f(x)=x + 1/x ` in the interval [1, 3]
उत्तर
`f(x)=x + 1/x ` x ∈ [1, 3]
(i) f (x) is continuous for x ∈ [1,3]
(ii) f (x) is differentiable for x ∈ (1,3)
∴ LMVT is applicable.
`f(1)=1+1/1 = 2 and f(3)= 3 + 1/3 = 10/3 `
`f'(x)= 1- 1/x^2 therefore f'(c)= 1-1/c^2`
`therefore f'(c)= (f(b)-f(a))/(b - a) rArr 1-(1)/c^2= (10/3 -2)/(3-1) = ((10-6)/3)/2 = 4/6=2/3`
`rArr 1- 2/3 = 1/c^2 rArr 1/3=1/c^2 ∴ c^2 = 3`
∴ c = ± `sqrt3 but c=sqrt3 ∈ [1,3]`
LMVT is vertified.
APPEARS IN
संबंधित प्रश्न
Verify Rolle's theorem for the function
f(x)=x2-5x+9 on [1,4]
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.
Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
Verify Lagrange's Mean Value Theorem for the following function:
`f(x ) = 2 sin x + sin 2x " on " [0, pi]`
The value of c in Rolle’s Theorem for the function f(x) = e x sinx, x ∈ π [0, π] is ______.
f(x) = `sin^4x + cos^4x` in `[0, pi/2]`
f(x) = log(x2 + 2) – log3 in [–1, 1]
f(x) = `sqrt(4 - x^2)` in [– 2, 2]
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) = x3 – 2x2 – x + 3 in [0, 1]
f(x) = `sqrt(25 - x^2)` in [1, 5]
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
The value of c in Rolle’s theorem for the function f(x) = x3 – 3x in the interval `[0, sqrt(3)]` is ______.
Rolle’s theorem is applicable for the function f(x) = |x – 1| in [0, 2].
The value of c in Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]` is ____________.
A value of c for which the Mean value theorem holds for the function f(x) = logex on the interval [1, 3] is ____________.
The value of c in mean value theorem for the function f(x) = (x - 3)(x - 6)(x - 9) in [3, 5] is ____________.
If the greatest height attained by a projectile be equal to the horizontal range, then the angle of projection is
If A, G, H are arithmetic, geometric and harmonic means between a and b respectively, then A, G, H are
Let a function f: R→R be defined as
f(x) = `{(sinx - e^x",", if x < 0),(a + [-x]",", if 0 < x < 1),(2x - b",", if x > 1):}`
where [x] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) 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 ______.
Let f(1) = –2 and f'(x) ≥ 4.2 for 1 ≤ x ≤ 6. The possible value of f(6) lies in the interval ______.
