Chapter 15 Mean Value Theorems Ex 15.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 2(i)

Solution 2(i)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 2(vii)

Solution 2(vii)

Question 2(viii)

Solution 2(viii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Here,

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 3(x)

Solution 3(x)

Question 3(xi)

Solution 3(xi)

Question 3(xii)

Solution 3(xii)

Question 3(xiii)

Solution 3(xiii)

Question 3(xiv)

Solution 3(xiv)

Question 3(xv)

Solution 3(xv)

Question 3(xvi)

Solution 3(xvi)

Question 3(xvii)

Solution 3(xvii)

Question 3(xviii)

Verify Rolle’s theorem for function f(x) = sin x – sin 2x on [0, pi] on the indicated intervals.Solution 3(xviii)

Question 7

Solution 7

x = 0 then y = 16

Therefore, the point on the curve is (0, 16) Question 8(i)

Solution 8(i)

x = 0, then y = 0

Therefore, the point is (0, 0)Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

x = 1/2, then y = – 27

Therefore, the point is (1/2, – 27)Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 2(ii)

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

Solution 2(ii)

Given function is 

As the given function is a polynomial, so it is continuous and differentiable everywhere.

Let’s find the extreme values

Therefore, f(2) = f(6).

So, Rolle’s theorem is applicable for f on [2, 6].

Let’s find the derivative of f(x)

Take f'(x) = 0

As 4 ∈ [2, 6] and f'(4) = 0.

Thus, Rolle’s theorem is verified.

Chapter 15 Mean Value Theorems Ex. 15.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(x)

Solution 1(x)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 1(xiii)

Solution 1(xiii)

Question 1(xiv)

Solution 1(xiv)

Question 1(xv)

Solution 1(xv)

Question 1(xvi)

Solution 1(xvi)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11