Advertisements
Advertisements
Question
f(x) = `x(x + 3)e^((–x)/2)` in [–3, 0]
Solution
We have, f(x) = `x(x + 3)e^((–x)/2)`
Since polynomial function x(x + 3) and exponential function `"e"^((-x)/2)` are continuous and differentiable in R, given function f(x) is also continuous and differentiable in R
Also f(0) = f(–3) = 0
So, conditions of Rolle's theorem are satisfied.
Hence, there exists a real number c ∈ (–3, 0) such that f'(c) = 0
Now f(x) = `(x^2 + 3x)"e"^((-x)/2)`
∴ f'(x) = `(2x + 3)"e"^((-x)/2) - 1/2 "e"^((-x)/2) (x^2 + 3x)`
= `- 1/2 "e"^((-x)/2) (x^2 + 3x - 4x - 6)`
= `-1/2 "e"^((-x)/2)(x^2 - x - 6)`
So, f'(x) = 0
⇒ `- 1/2 "e"^((-x)/2) ("c" + 2)("c" - 3)` = 0
⇒ c = –2 ∈ (–3, 0)
Therefore, Rolle's theorem has been verified.
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]
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 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]`
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]`.
f(x) = x(x – 1)2 in [0, 1]
f(x) = `sin^4x + cos^4x` in `[0, pi/2]`
f(x) = x3 – 2x2 – x + 3 in [0, 1]
f(x) = sinx – sin2x in [0, π]
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
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 ____________.
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
Rolle's Theorem holds for the function x3 + bx2 + cx, 1 ≤ x ≤ 2 at the point `4/3`, the value of b and c 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 ______.
`lim_(x→0) sqrt(1 - cosx)/(sqrt(2)x)` is ______.
Let f(1) = –2 and f'(x) ≥ 4.2 for 1 ≤ x ≤ 6. The possible value of f(6) lies in the interval ______.
