Advertisements
Advertisements
प्रश्न
f(x) = `sqrt(4 - x^2)` in [– 2, 2]
उत्तर
We have, `sqrt(4 - x^2) = (4 - x^2)^(1/2)`
Since (4 – x2) and square root function are continuous and differentiable in their domain, given function f(x) is also continuous and differentiable in [– 2, 2]
Also f(–2) = f(2) = 0
So, conditions of Rolle's theorem are satisfied.
Hence, there exists a real number c ∈ (–2, 2) such that f'(c) = 0.
Now f'(x) = `1/2(4 - x^2)^((-1)/2)(-2x)`
= `- x/sqrt(4 - x^2)`
So, f'(c) = 0
⇒ `"c"/sqrt(4 - "c"^2)` = 0
⇒ c = 0 ∈ (–2, 2)
Hence Rolle's theorem has been verfired.
APPEARS IN
संबंधित प्रश्न
Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]
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]
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 ∈ [– 2, 2]
Verify Mean Value Theorem, if f (x) = x2 – 4x – 3 in the interval [a, b], where a = 1 and b = 4.
Verify Rolle’s theorem for the following function:
`f(x) = e^(-x) sinx " on" [0, pi]`
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 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 Mean value theorem for the function f(x) = 2sin x + sin 2x on [0, π].
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 – 1)2 in [0, 1]
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):}`
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]
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].
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 ____________.
`lim_(x→0) sqrt(1 - cosx)/(sqrt(2)x)` is ______.
