Advertisements
Advertisements
प्रश्न
f(x) = log(x2 + 2) – log3 in [–1, 1]
उत्तर
We have, f(x) = log(x2 + 2) – log3
We know that x2 + 2 and logarithmic function are continuous and differentiable
∴ f(x) = log(x2 + 2) – log3 is also continuous and differentiable.
Now f(–1) = f(1) = log3 - log3 = 0
So, conditions of Rolle's theorem are satisfied.
Hence, there exists atleast one c ∈ (–1, 1) such that f'(c) = 0
f(x) = `(2"c")/("c"^2 + 2) - 0` = 0
⇒ c = 0 ∈ (–1, 1)
Hence, Rolle's theorem has been verified.
APPEARS IN
संबंधित प्रश्न
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 + 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) = x2 – 1 for x ∈ [1, 2]
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.
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 Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]`.
Verify mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5].
The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is ______.
f(x) = `x(x + 3)e^((–x)/2)` in [–3, 0]
Find the points on the curve y = (cosx – 1) in [0, 2π], where the tangent is parallel to x-axis
f(x) = `1/(4x - 1)` in [1, 4]
f(x) = x3 – 2x2 – x + 3 in [0, 1]
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)
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 ______.
If x2 + y2 = 1, then ____________.
A value of c for which the Mean value theorem holds for the function f(x) = logex on the interval [1, 3] is ____________.
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 ______.
Let f(1) = –2 and f'(x) ≥ 4.2 for 1 ≤ x ≤ 6. The possible value of f(6) lies in the interval ______.
