Advertisements
Advertisements
प्रश्न
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
उत्तर
We have, y = 2x2 – 5x + 3, which is polynomial function.
So it is continuous and differentiable.
Thus conditions of mean value theorem are satisfied.
Hence, there exists atleast one c ∈ (1, 2) such that,
f'(c) = `("f"(2) - "f"(1))/(2 - 1)`
⇒ 4c – 5 = `(1 - 0)/1`
⇒ 4c – 5 = 1
∴ c = `3/2 ∈ (1, 2)`
For x = `3/2`, y = `2(3/2)^2 - 5(3/2) + 3` = 0
Hence, `(3/2, 0)` is the points 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.
APPEARS IN
संबंधित प्रश्न
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]
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]
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) = e^(-x) sinx " on" [0, pi]`
Verify Lagrange's Mean Value Theorem for the following function:
`f(x ) = 2 sin x + sin 2x " 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 Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]`.
The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is ______.
f(x) = `sin^4x + cos^4x` in `[0, pi/2]`
f(x) = log(x2 + 2) – log3 in [–1, 1]
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]
For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.
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 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 ______.
Let f(1) = –2 and f'(x) ≥ 4.2 for 1 ≤ x ≤ 6. The possible value of f(6) lies in the interval ______.
