Advertisements
Advertisements
Question
Verify Rolle’s theorem for the following function:
`f(x) = e^(-x) sinx " on" [0, pi]`
Solution
`f(x) = e^(-x). sin x`
1) f (x) is continuous on `[0,pi]` because `e^(-x)` and sin x are continuous function on its domain
2) `e^(-x)` and sin x is differentiable on `(0, pi)`
3) `f(0) = e^(-0). sin 0 = 0`
`f(pi) = e^(-pi). sin pi = 0`
4) Let c be number such that f'(x) = 0
`:. f'(x) = e^(-x). cos x _+ sin x. e^(-x) (-1)`
`:. f'(c) = e^(-c) (cos c - sin c)`
`:. f'(c) = 0`
`e^(-c) (cos c - sin c) = 0`
`:.e^(-c) = 0`
`:. cos C - sin C = 0
tan C =1
`:. C = pi/4, (3pi)/4 , (5pi)/4,.....`
`:. pi/4 in [0, pi]`
∴ Rolle's theorem verify
APPEARS IN
RELATED QUESTIONS
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.
Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
Verify the Lagrange’s mean value theorem for the function:
`f(x)=x + 1/x ` in the interval [1, 3]
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 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) = `x(x + 3)e^((–x)/2)` in [–3, 0]
f(x) = `sqrt(4 - x^2)` in [– 2, 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) = `1/(4x - 1)` in [1, 4]
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)
For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.
If x2 + y2 = 1, then ____________.
The value of c in Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]` 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
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 ______.
