Advertisements
Advertisements
Question
Check the validity of the Rolle’s theorem for the following functions : f(x) = 2x2 – 5x + 3, x ∈ [1, 3].
Solution
The function f given as f(x) = 2x2 – 5x + 3 is a polynominal function. Hence, it is continuous on [1, 3] and differentiable on (1, 3).
Now,
f(1)
= 2(1)2 – 5(1) + 3
= 2 – 5 + 3
= 0
and
f(3)
= 2(3)2 – 5(3) + 3
= 18 – 15 + 3
= 3 + 3
= 6
∴ f(1) ≠ f(3)
Hence, the conditions of the Rolle's theorem are not satisfied.
APPEARS IN
RELATED QUESTIONS
Check the validity of the Rolle’s theorem for the following functions : f(x) = x2 – 4x + 3, x ∈ [1, 3]
Check the validity of the Rolle’s theorem for the following functions : f(x) = e–x sin x, x ∈ [0, π].
Check the validity of the Rolle’s theorem for the following functions : f(x) = sin x – cos x + 3, x ∈ [0, 2π].
Check the validity of the Rolle’s theorem for the following function:
f(x) = x2, if 0 ≤ x ≤ 2
= 6 – x, if 2 < x ≤ 6.
Check the validity of the Rolle’s theorem for the following function:
f(x) = `x^(2/3), x ∈ [ - 1, 1]`
Given an interval [a, b] that satisfies hypothesis of Rolle's theorem for the function f(x) = x4 + x2 – 2. It is known that a = – 1. Find the value of b.
Verify Rolle’s theorem for the following functions : f(x) = sin x + cos x + 7, x ∈ [0, 2π]
Verify Rolle’s theorem for the following functions : f(x) = `sin(x/2), x ∈ [0, 2pi]`
If Rolle's theorem holds for the function f(x) = x3 + px2 + qx + 5, x ∈ [1, 3] with c = `2 + (1)/sqrt(3)`, find the values of p and q.
If Rolle’s theorem holds for the function f(x) = (x –2) log x, x ∈ [1, 2], show that the equation x log x = 2 – x is satisfied by at least one value of x in (1, 2).
The function f(x) = `x(x + 3)e^(-(x)/2)` satisfies all the conditions of Rolle's theorem on [– 3, 0]. Find the value of c such that f'(c) = 0.
Choose the correct option from the given alternatives :
If the function f(x) = ax3 + bx2 + 11x – 6 satisfies conditions of Rolle's theoreem in [1, 3] and `f'(2 + 1/sqrt(3))` = 0, then values of a and b are respectively
Verify Rolle’s theorem for the function f(x) `(2)/(e^x + e^-x)` on [– 1, 1].
A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 1.5 m /sec. The length of the higher point of the when foot of the ladder is 4 m away from the wall decreases at the rate of ______
lf Rolle's theorem for f(x) = ex (sin x – cos x) is verified on `[π/4, (5π)/4]`, then the value of c is ______.
If Rolle's theorem holds for the function f(x) = x3 + bx2 + ax + 5, x ∈ [1, 3] with c = `2 + 1/sqrt(3)`, find the values of a and b.
