Advertisements
Advertisements
प्रश्न
f(x) = sinx – sin2x in [0, π]
उत्तर
We have, f(x) = sinx – sin2x in [0, π]
We know that all trigonometric functions are continuous and differentiable their domain, given function is also continuous and differentiable
So, condition of mean value theorem are satisfied.
Hence, there exists atleast one c ∈ (0, π) such that,
f'(c) = `("f"(pi) - "f"(0))/(pi - 0)`
⇒ cos c – 2 cos 2c = `(sin pi - sin 2pi - sin 0 + sin 0)/(pi - 0)`
⇒ 2 cos 2c – cos c = 0
⇒ 2(2 cos2c – 1) – cos c = 0
⇒ 4cos2c – cos c – 2 = 0
⇒ cos c = `(1 +- sqrt(1 + 32))/8`
= `(1 +- sqrt(33))/8`
⇒ c = `cos^-1 ((1 +- sqrt(33))/8) ∈ (0, π)`
Hence, mean value theorem has been verified.
APPEARS IN
संबंधित प्रश्न
Verify Rolle’s theorem for the function f (x) = x2 + 2x – 8, x ∈ [– 4, 2].
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]
If f : [– 5, 5] → R is a differentiable function and if f ′(x) does not vanish anywhere, then prove that f (– 5) ≠ f (5).
Verify Rolle’s theorem for the following function:
f (x) = x2 - 4x + 10 on [0, 4]
Verify Lagrange's Mean Value Theorem for the following function:
`f(x ) = 2 sin x + sin 2x " on " [0, pi]`
f(x) = (x-1)(x-2)(x-3) , x ε[0,4], find if 'c' LMVT can be applied
Verify the Lagrange’s mean value theorem for the function:
`f(x)=x + 1/x ` in the interval [1, 3]
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 mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5].
The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is ______.
f(x) = x(x – 1)2 in [0, 1]
f(x) = log(x2 + 2) – log3 in [–1, 1]
f(x) = `sqrt(4 - x^2)` in [– 2, 2]
f(x) = x3 – 2x2 – x + 3 in [0, 1]
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
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].
If x2 + y2 = 1, then ____________.
If A, G, H are arithmetic, geometric and harmonic means between a and b respectively, then A, G, H are
Value of' 'c' of the mean value theorem for the function `f(x) = x(x - 2)`, when a = 0, b = 3/2, is
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 ______.
