Advertisements
Advertisements
Question
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):}`
Solution
We have, f(x) = `{{:(x^2 + 1",", "if" 0 ≤ x ≤ 1),(3 - x",", "if" 1 ≤ x ≤ 2):}`
We know that polynomial function is everywhere continuous and differentiability.
So, f(x) is continuous and differentiable at all points except possibly at x = 1.
Now `lim_(x -> 1^-) (x^2 + 1)` = 1 + 1 = 2
And `lim_(x -> 1^+) (3 - x)` = 3 – 1 = 2
Also f(1) = 12 + 1 = 2
So, f(x) is continuous at x = 1
Also f'(x) = `{{:(2x",", "if" 0 < x < 1),(-x",", "if" 1 < x 2):}`
f'(1) = 2(1) = 2
And f'(1) = –1
Thus f'(1) ≠ f'(1).
So, f(x) is not differentiable at x = 1
Hence, Rolle's theorem is not applicable on the interval [0, 2].
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]
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 Rolle’s theorem for the following function:
`f(x) = e^(-x) sinx " 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) = 2sin x + sin 2x on [0, π].
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) = `x(x + 3)e^((–x)/2)` in [–3, 0]
f(x) = `1/(4x - 1)` in [1, 4]
f(x) = x3 – 2x2 – x + 3 in [0, 1]
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) = 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 ____________.
If `1/(a + ω) + 1/(b + ω) + 1/(c + ω) + 1/(d + ω) = 1/ω`, where a, b, c, d ∈ R and ω is a cube root of unity then `sum 3/(a^2 - a + 1)` is equal to
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 ______.
