## Chapter 15 – Mean Value Theorems Exercise Ex. 15.1

Here,

Verify Rolle’s theorem for function *f(x)* = sin *x* – sin 2*x* on [0, p]

on the indicated intervals.

x = 0 then y = 16

Therefore, the point on the curve is (0, 16)

x = 0, then y = 0

Therefore, the point is (0, 0)

x = 1/2, then y = – 27

Therefore, the point is (1/2, – 27)

## Chapter 15 – Mean Value Theorems Exercise Ex. 15.2

## Chapter 15 – Mean Value Theorems Exercise Ex. 15VSAQ

## Chapter 15 – Mean Value Theorems Exercise MCQ

If the polynomial equation a_{0}x^{n} + a_{n-1}x^{n}^{ – 1} + a_{n – 2}x^{n}^{ – 2} + ……+ a_{2}x^{2} + a_{1}x + a_{0} = 0 n being a positive integer, has two different real roots α and β, the equation n a_{n}x^{n-1} + (n – 1)a_{n}_{ – 1}x^{n}^{ – 2} + …. + a_{1} = 0 has

- exactly one root
- almost one root
- at least one root
- no root

Correct option: (c)

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

- (0, 1)
- (1, 2)
- (0, 2)
- none of these

Correct option: (c)

- 1
- 2
- none of these

Correct option: (b)

- a < x
_{1}≤ b - a ≤ x
_{1}< b - a < x
_{1}< b - a ≤ x
_{1}≤ b

Correct option: (c)

Using statement of Lagrange’s mean value theorem function is continuous on [a,b], differentiable on (a,b) then there exists c such that a < x_{1}< b.

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

- any interval
- the interval [0, π]
- the interval (0, π/2)
- none of these

Correct option: (b)

ϕ(x) is continuous and differentiable function then using statement of Rolle’s theorem f(a)=f(b). Hence, here sin 0=0 also sin п=0. The answer is [0, ].

The value of c in Rolle’s theorem when f(x) = 2x^{3} – 5x^{2} – 4x + 3, is x ∈ [1/3, 3]

- 2
- -1/3
- -2
- 2/3

Correct option: (a)

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

- e
^{1/1 – e} - e
^{(e – 1)(2e – 1)}

Correct option: (a)

Correct answer: (c)

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

- 1
- 1/2
- 2/3
- 3/2

Correct option: (d)

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

- 1
- -1
- 3/2
- 1/3

Correct option: (a)

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

- π/6
- π/4
- π/2
- 3π/4

Correct option: (d)