###### Advertisements

###### Advertisements

Verify Mean value theorem for the function f(x) = 2sin x + sin 2x on [0, π].

###### Advertisements

#### Solution

f(x) = 2sin x + sin 2x on [0, π].

f(x) is continuous in [0, π].

f(x) is differentiable in [0, π].

∴ Mean value theorem is applieable

f(0) = 0, f(x) = 0

f'(x) = 2 cos x + 2 cos 2x

f'(c) = 2 cos c + 2 cos 2c

f'(c) = `(f(pi) - f(0))/(pi - 0) = 0`

∴ 2 cos c + 2 cos 2c = 0

⇒ (2 cos c - 1)(cos c + 1) = 0

⇒ cos c`1/2`

⇒ c = `pi/3` ∈ (0, π)

Hence mean value theorem is 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)=x^{2}-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]

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) = [x] for x ∈ [– 2, 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^{2} – 1 for x ∈ [1, 2]

Verify Mean Value Theorem, if f (x) = x^{2} – 4x – 3 in the interval [a, b], where a = 1 and b = 4.

Verify Mean Value Theorem, if f (x) = x^{3} – 5x^{2} – 3x in the interval [a, b], where a = 1 and b = 3. Find all c ∈ (1, 3) for which f ′(c) = 0.

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.

Verify Rolle’s theorem for the following function:

f (x) = x^{2} - 4x + 10 on [0, 4]

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) = e^{x} (sin x – cos x) on `[ (π)/(4), (5π)/(4)]`.

The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is ______.

f(x) = x(x – 1)^{2} in [0, 1]

f(x) = `sin^4x + cos^4x` in `[0, pi/2]`

f(x) = log(x^{2} + 2) – log3 in [–1, 1]

f(x) = `x(x + 3)e^((–x)/2)` in [–3, 0]

f(x) = `sqrt(4 - x^2)` in [– 2, 2]

Using Rolle’s theorem, find the point on the curve y = x(x – 4), x ∈ [0, 4], where the tangent is parallel to x-axis

f(x) = `1/(4x - 1)` in [1, 4]

f(x) = x^{3} – 2x^{2} – 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)

For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.

Rolle’s theorem is applicable for the function f(x) = |x – 1| in [0, 2].

If x^{2} + y^{2} = 1, then ____________.

A value of c for which the Mean value theorem holds for the function f(x) = log_{e}x on the interval [1, 3] is ____________.

The value of c in mean value theorem for the function f(x) = (x - 3)(x - 6)(x - 9) in [3, 5] 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

Value of' 'c' of the mean value theorem for the function `f(x) = x(x - 2)`, when a = 0, b = 3/2, 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 x^{3} + bx^{2} + 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 ______.

P(x) be a polynomial satisfying P(x) – 2P'(x) = 3x^{3} – 27x^{2} + 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 ______.

`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 ______.