Exercise 6.2 Solutions
MONOTONICITY • INCREASING AND DECREASING FUNCTIONS
💡 Test for Monotonicity
A function $f(x)$ is Increasing on an interval if $f'(x) > 0$ and Decreasing if $f'(x) < 0$.
Question 01
Show that $f(x)=3x+17$ is increasing on $\mathbb{R}$.
$f'(x) = \frac{d}{dx}(3x+17) = 3$.
Since $3 > 0$ for all $x \in \mathbb{R}$, $f'(x) > 0$.
Since $3 > 0$ for all $x \in \mathbb{R}$, $f'(x) > 0$.
Hence, $f(x)$ is increasing on $\mathbb{R}$.
Question 02
Show that $f(x)=e^{2x}$ is increasing on $\mathbb{R}$.
$f'(x) = 2e^{2x}$.
We know that $e^{2x} > 0$ for any real value of $x$. Thus, $f'(x) > 0$ for all $x \in \mathbb{R}$.
We know that $e^{2x} > 0$ for any real value of $x$. Thus, $f'(x) > 0$ for all $x \in \mathbb{R}$.
Hence, $f(x)$ is increasing on $\mathbb{R}$.
Question 03
Show that $f(x)=\sin x$ is (a) increasing in $(0, \pi/2)$ (b) decreasing in $(\pi/2, \pi)$.
$f'(x) = \cos x$.
(a) In $(0, \pi/2)$, $\cos x > 0 \implies f'(x) > 0$. (Increasing)
(b) In $(\pi/2, \pi)$, $\cos x < 0 \implies f'(x) < 0$. (Decreasing)
(c) In $(0, \pi)$, $f'(x)$ changes sign, so it is neither increasing nor decreasing.
Question 04
Find intervals where $f(x)=2x^2-3x$ is increasing and decreasing.
$f'(x) = 4x – 3$. Set $f'(x) = 0 \implies x = 3/4$.
Question 05
Find intervals for $f(x)=2x^3-3x^2-36x+7$.
$f'(x) = 6x^2 – 6x – 36 = 6(x-3)(x+2)$. Critical points: $-2, 3$.
| Interval | Sign of $f'(x)$ | Nature |
|---|---|---|
| $(-\infty, -2)$ | $(+) \cdot (-) \cdot (-) = +$ | Increasing |
| $(-2, 3)$ | $(+) \cdot (-) \cdot (+) = -$ | Decreasing |
| $(3, \infty)$ | $(+) \cdot (+) \cdot (+) = +$ | Increasing |
Question 06
Find intervals for strictly increasing/decreasing:
(a) $x^2+2x-5$: $f’=2(x+1)$. Inc: $(-1, \infty)$, Dec: $(-\infty, -1)$.
(b) $10-6x-2x^2$: $f’=-2(2x+3)$. Inc: $(-\infty, -3/2)$, Dec: $(-3/2, \infty)$.
(c) $-2x^3-9x^2-12x+1$: $f’=-6(x+1)(x+2)$. Inc: $(-2, -1)$.
(e) $(x+1)^3(x-3)^3$: $f’=6(x+1)^2(x-3)^2(x-1)$. Inc: $x > 1$, Dec: $x < 1$.
Question 07
Show $y = \frac{2\log(1+x)-x}{1+x}$ is increasing for $x > -1$.
$\frac{dy}{dx} = \frac{x}{(1+x)^2}$. Since $(1+x)^2 > 0$ and $x/(1+x)^2 > 0$ for $x > 0$.
(Correcting text version): $\frac{dy}{dx} = \frac{x^2}{(1+x)^2} \ge 0$ for all $x > -1$.
(Correcting text version): $\frac{dy}{dx} = \frac{x^2}{(1+x)^2} \ge 0$ for all $x > -1$.
Hence Proved
Question 08
Find values of $x$ where $y = [x(x-2)]^2$ is increasing.
$y’ = 2x(x-2)(2x-2) = 4x(x-1)(x-2)$. Critical points: $0, 1, 2$.
Question 09
Prove $y = \frac{4\sin\theta}{2+\cos\theta} – \theta$ is increasing in $[0, \pi/2]$.
$\frac{dy}{d\theta} = \frac{\cos\theta(4-\cos\theta)}{(2+\cos\theta)^2}$.
In first quadrant, $\cos\theta > 0$ and $(4-\cos\theta) > 0$. So $\frac{dy}{d\theta} > 0$.
In first quadrant, $\cos\theta > 0$ and $(4-\cos\theta) > 0$. So $\frac{dy}{d\theta} > 0$.
Strictly Increasing
Question 10
Prove logarithmic function is increasing on $(0, \infty)$.
$f(x) = \log x \implies f'(x) = 1/x$.
For $x > 0$, $1/x$ is always positive.
For $x > 0$, $1/x$ is always positive.
Increasing on $(0, \infty)$
Question 11
Prove $x^2-x+1$ is neither increasing nor decreasing on $(-1, 1)$.
$f'(x) = 2x – 1$. Root at $1/2$.
On $(-1, 1/2)$, $f’ < 0$ (Dec). On $(1/2, 1)$, $f' > 0$ (Inc).
On $(-1, 1/2)$, $f’ < 0$ (Dec). On $(1/2, 1)$, $f' > 0$ (Inc).
Neither increasing nor decreasing
Question 12
Which functions decrease on $(0, \pi/2)$?
$\cos x$, $\cos 2x$, and $\cos 3x$ decrease as their derivatives are negative in specific parts of the interval.
Correct: A, B, C
Question 13
Interval for $f(x)=x^{100}+\sin x – 1$ decreasing.
$f'(x) = 100x^{99} + \cos x$. For intervals like $(0, 1)$, $f’ > 0$.
Result: None of these (Option D)
Question 14
Find $a$ if $f(x)=x^2+ax+1$ is increasing on $[1, 2]$.
$f'(x) = 2x + a \ge 0$. Min value at $x=1 \implies 2+a \ge 0$.
Result: $a \ge -2$
Question 15
Show $f(x)=x+1/x$ increases on $I$ disjoint from $[-1, 1]$.
$f'(x) = 1 – 1/x^2$. If $x \notin [-1, 1]$, $x^2 > 1 \implies 1/x^2 < 1$.
So $1 – 1/x^2 > 0$.
So $1 – 1/x^2 > 0$.
Hence Proved
Question 16
Prove $\log \sin x$ is inc on $(0, \pi/2)$ and dec on $(\pi/2, \pi)$.
$f'(x) = \cot x$.
In Q1, $\cot x > 0$. In Q2, $\cot x < 0$.
In Q1, $\cot x > 0$. In Q2, $\cot x < 0$.
Verified
Question 17
Prove $\log |\cos x|$ is dec on $(0, \pi/2)$ and inc on $(3\pi/2, 2\pi)$.
$f'(x) = -\tan x$.
In Q1, $\tan x > 0 \implies f’ < 0$. In Q4, $\tan x < 0 \implies f' > 0$.
In Q1, $\tan x > 0 \implies f’ < 0$. In Q4, $\tan x < 0 \implies f' > 0$.
Verified
Question 18
Prove $x^3-3x^2+3x-100$ is increasing on $\mathbb{R}$.
$f'(x) = 3(x-1)^2$.
Since a square is always $\ge 0$, $f'(x) \ge 0$.
Since a square is always $\ge 0$, $f'(x) \ge 0$.
Increasing on R
Question 19
Interval where $y = x^2 e^{-x}$ increases.
$y’ = xe^{-x}(2-x)$. Positive when $x(2-x) > 0$.
Correct: (0, 2) (Option D)