Table of Contents
Chapter 17 Increasing and Decreasing Functions Ex. 17.1
Question 1
Solution 1
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
Chapter 17 Increasing and Decreasing Functions Ex. 17.2
Question 1(i)
Find the intervals in which the following functions are increasing or decreasing:
10 – 6x – 2x2Solution 1(i)
Question 1(ii)
Find the intervals in which the following functions are increasing or decreasing:
x2 + 2x – 5Solution 1(ii)
Question 1(iii)
Find the intervals in which the following functions are increasing or decreasing:
6 – 9x – x2Solution 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(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 1(xvii)
Solution 1(xvii)
Question 1(xviii)
Solution 1(xviii)
Question 1(xix)
Solution 1(xix)
Question 1(xx)
Solution 1(xx)
Question 1(xxi)
Solution 1(xxi)
Question 1(xxii)
Solution 1(xxii)
Question 1(xxiii)
Solution 1(xxiii)
Question 1(xxiv)
Solution 1(xxiv)
Question 1(xxv)
Find the values of x for which the function y = [x(x – 2)]2 is increasing or decreasingSolution 1(xxv)
Question 1(xxvi)
Find the interval in which the following function is increasing or decreasing.
f(x) = 3x4– 4x3– 12x2 + 5Solution 1(xxvi)
Question 1(xxvii)
Find the interval in which the following function is increasing or decreasing.
Solution 1(xxvii)
Question 1(xxviii)
Find the interval in which the following function is increasing or decreasing.
Solution 1(xxviii)
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Show that the function given by f(x) = sin x is
(a) increasing in (0, π/2)
(b) decreasing in (π/2, π)
(c) neither increasing nor decreasing in (0, π)Solution 7
Question 8
Prove that the function f given by f(x) = log sin x is increasing on and decreasing on Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30(i)
Solution 30(i)
Question 31
Solution 31
Question 32
Solution 32
Question 33
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π)Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39(i)
Find the interval in which f(x) is increasing or decreasing:
Solution 39(i)
Question 39(ii)
Find the interval in which f(x) is increasing or decreasing:
Solution 39(ii)
Question 39(iii)
Find the interval in which f(x) is increasing or decreasing:
Solution 39(iii)
Question 1(x)
Find the intervals in which the following functions are increasing or decreasing:
Solution 1(x)
Given:
Differentiating w.r.t x, we get
Take f'(x) = 0
Clearly, f'(x) > 0 if x < -2 or x > -1
And, f'(x) < 0 if -2 < x < -1
Thus, f(x) increases on and decreases on Question 1(xxix)
Find the intervals in which the following functions are increasing or decreasing:
Solution 1(xxix)
Given:
Differentiating w.r.t x, we get
Take f'(x) = 0
The points x = 2, 4 and -3 divide the number line into four disjoint intervals namely
Consider the interval
In this case, x – 2 < 0, x – 4 < 0 and x + 3 < 0
Therefore, f'(x) < 0 when
Thus the function is decreasing in
Consider the interval
In this case, x – 2 < 0, x – 4 < 0 and x + 3 > 0
Therefore, f'(x) > 0 when
Thus the function is increasing in
Now, consider the interval
In this case, x – 2 > 0, x – 4 < 0 and x + 3 > 0
Therefore, f'(x) < 0 when
Thus the function is decreasing in
And now, consider the interval
In this case, x – 2 > 0, x – 4 > 0 and x + 3 > 0
Therefore, f'(x) < 0 when
Thus the function is increasing in Question 30(ii)
Prove that the following function is increasing on R:
Solution 30(ii)
Given:
Differentiating w.r.t x, we get
Now,
Hence, f(x) is an increasing function for all x.
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