Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

f(x) = 3 + (x – 2)^2/3 on [1, 3]

Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

f(x = [x] for – 1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x

Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

f(x) = sin 1/x for – 1 ≤ x ≤ 1

Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

f(x) = 2x² – 5x + 3 on [1, 3]

Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

f(x) = x^2/3 on [ – 1, 1]

Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x² – 8x + 12 on [2, 6]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x² – 4x + 3 on [1, 3]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = (x – 1) (x – 2)² on [1, 2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x(x – 1)² on [0, 1]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = (x² – 1)(x – 2) on [ – 1, 2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x(x – 4)² on [0, 4]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x(x – 2)² on [0, 2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x² + 5x + 6 on [ – 3, – 2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = cos 2 (x – π/4) on [0, π/2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = sin 2x on [0, π/2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = cos 2x on [– π/4, π/4]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = e^x sin x on [0, π]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = e^x cos x on [– π/2, π/2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = cos 2x on [0, π]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = sin 3x on [0, π]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = log(x² + 2) – log 3 on [–1, 1]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = sin x + cos x on [0, π/2]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = 2 sin x + sin 2x on [0, π]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = 4^{sin x} on [0, π]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x² – 5x + 4 on [1, 4]

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = sin x – sin 2x on [0, π]

Using Rolle’s theorem, find points on the curve y = 16 – x², xϵ[ – 1,1], where the tangent is parallel to the x – axis.

At what points on the following curves, is the tangent parallel to the x–axis?

y = x² on [–2, 2]

At what points on the following curves, is the tangent parallel to the x–axis?

on [–1, 1]

At what points on the following curves, is the tangent parallel to the x–axis?

y = 12(x + 1) (x – 2) on [–1, 2]

If f:[ – 5,5]→R is differentiable and if f’(x) doesn’t vanish anywhere, then prove that f( – 5)≠f(5).

Examine if the Rolle’s theorem applies to anyone of the following functions:

f(x) = [x] for xϵ[5,9]

Examine if the Rolle’s theorem applies to anyone of the following functions:

f(x) = [x] for xϵ[ – 2,2]

It is given that the Rolle’s theorem holds for the function f(x) = x³ + bx² + cx, xϵ[1,2] at the point x = 4/3. Find the values of b and c.

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x² – 1 on [2, 3]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x³ – 2x² – x + 3 on [0, 1]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x (x – 1) on [1, 2]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x² – 3x + 2 on [ – 1, 2]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = 2x² – 3x + 1 on [1, 3]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x² – 2x + 4 on [1, 5]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = 2x – x² on [0, 1]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = (x – 1) (x – 2)(x – 3) on [0, 4]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = tan ^{– 1} x on [0, 1]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x (x + 4)² on [0, 4]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x² + x – 1 on [0, 4]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = sin x – sin 2x – x on [0, π]

Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :

f(x) = x³ – 5x² – 3x on [1, 3]

Discuss the applicability of Lagrange’s mean value theorem for the function f(x) = |x| on [ – 1, 1].

Show that the Lagrange's mean value theorem is not applicable to the function f(x) = 1/x on [–1, 1].

"Verify the hypothesis and conclusion of Lagrange’s mean value theorem for the function

Find a point on the parabola y = (x – 4)², where the tangent is parallel to the chord joining (4, 0) and (5, 1).

Find a point on the curve y = x² + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2).

Find a point on the parabola y = (x – 3)², where the tangent is parallel to the chord joining (3, 0) and (4, 1).

Find points on the curve y = x³ – 3x, where the tangent to the curve is parallel to the chord joining (1, – 2) and (2, 2).

Find a point on the curve y = x³ + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28).

Let C be a curve defined parametrically as x = a cos³ θ, y = a sin³ θ, 0 ≤θ ≤ π/2. Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).

Using Lagrange’s mean value theorem, prove that

(b – a ) sec² a < tan b – tan a < (b – a) sec² b, where 0 < a < b < π/2.

Mark the correct alternative in the following:

If the polynomial equation a₀xⁿ + a_n–1x^n–1 + a_n–2x^n–2 + …. a₂x² + a₁x + a₀ = 0

n being a positive integer, has two different real roots α and β, then between α and β, the equation n a_nx^n–1 + (n–1) a_n–1x^n–2 + … + a₁ = 0 has

Mark the correct alternative in the following:

If 4a + 2b + c = 0, then the equation 3ax² + 2bx + c = 0 has at least one real root lying in the interval.

Mark the correct alternative in the following:

For the function the value of c for the Lagrange’s mean value theorem is

Mark the correct alternative in the following:

If from Lagrange’s mean value theorem, we have then

Mark the correct alternative in the following:

Rolle’s theorem is applicable in case of ϕ (x) = a^sinx, a > 0 in

Mark the correct alternative in the following:

The value of e in Rolle’s theorem show f(x) = 2x³ – 5x² – 4x + 3, is x ϵ[1/3, 3]

Mark the correct alternative in the following:

`The value tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of is

Mark the correct alternative in the following:

The value of c in Rolle’s theorem for the function defined on [–1, 0] is

Mark the correct alternative in the following:

The value of c in Lagrange’ mean value theorem for the function f(x) = x (x – 2) where x ϵ [1, 2] is

Mark the correct alternative in the following:

The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval is

Mark the correct alternative in the following:

If f(x) = e^xsinx in [0, π], then c in Rolle’s theorem is

If f(x) = Ax² + Bx + C is such that f(a) = f(b), then write the value of c in Rolle’s theorem.

State Rolle’s theorem.

State Lagrange’s mean value theorem.

Find the value of c prescribed by Lagrange’s mean value theorem for the function defined on [2, 3].

What are the values of ‘a’ for which f(x) = a^x is increasing on R?

What are the values of ‘a’ for which f(x) = a^x is decreasing on R?

Write the set of values of ‘a’ for which f(x) = log_a x is increasing in its domain.

Write the set of values of ‘a’ for which f(x) = log_a x is decreasing in its domain.

Find ‘a’ for which f(x) = a (x + sinx) + a is increasing on R.

Find the vales of ‘a’ for which the function f(x) = sinx – ax + 4 is increasing function on R.

Find the set of values of ‘b’ for which f(x) = b(x + cosx) + 4 is decreasing on R.

Find the set of values of a’a’ for which f(x) = x + cosx + ax + b is increasing on R.

Write the set of values of k for which f(x) = kx – sinx is increasing on R.

If g(x) is a decreasing function on R and f(x) = tan^–1 {g(x)}. State whether f(x) is increasing or decreasing on R.

Write the set of values of a for which the function f(x) = ax + b is decreasing for all xϵR.

Write the interval in which f(x) = sin x + cos x, xϵ[0, π/2] is increasing.

State whether f(x) = tanx – x is increasing or decreasing its domain.

Write the set of values of a for which f(x) = cosx + a²x + b is strictly increasing on R.