Informative line

### Monotonicity

Learn Intervals in which the function is increasing or decreasing, Practice increasing and decreasing test & interval of decrease.

# Finding the Interval of Increase from Graph

• Consider the graph of a function $$f(x)$$ as shown.
• This shows the graph of an increasing (or going up or rising) function in the interval $$(a,\,b)$$.  • It can be observed that if we move towards $$b$$ from $$a$$, then the value of $$f(x)$$ increases along with the value of $$x.$$
• Thus, we can say that if $$f$$ is an increasing function in the interval $$(a,\,b)$$, then

$$f(x_2)>f(x_1)$$

whenever $$x_2>x_1\;\;\forall\;x_1,\,x_2\,\in(a,\,b)$$.  For example: Observe the given graph.

Here, the function $$y=f(x)$$ is increasing in the interval $$(b,\,c)$$.

In the interval $$(b,\,c)$$, as the value of $$x$$ is increasing in $$(b,\,c)$$, the value of $$f(x)$$ is also increasing.  Here, the function $$y=f(x)$$ is increasing in the interval $$(b,\,c)$$.

In the interval $$(b,\,c)$$, as the value of $$x$$ is increasing in $$(b,\,c)$$, the value of $$f(x)$$ is also increasing.

#### Observe the following graph and indicate the open intervals of increase of function $$'f'$$.

A $$f$$ is increasing in $$(a,b) \cup (c,e)$$

B $$f$$ is increasing in $$(b,c) \cup (d,e)$$

C $$f$$ is increasing in $$(b,d) \cup (d,e)$$

D $$f$$ is increasing in $$(a,b) \cup (c,d)$$

×

Observe that $$f$$ is decreasing (falling graph) in (a,b) , then increasing (rising graph) in (b,c) , decreasing  in (c,d) and increasing in (d,e) again.

$$\therefore$$ Interval of increase is $$(b,c)\cup(d,e)$$

Interval of decrease is $$(a,b)\cup(c,d)$$

Hence, option (B) is correct.

### Observe the following graph and indicate the open intervals of increase of function $$'f'$$. A

$$f$$ is increasing in $$(a,b) \cup (c,e)$$

.

B

$$f$$ is increasing in $$(b,c) \cup (d,e)$$

C

$$f$$ is increasing in $$(b,d) \cup (d,e)$$

D

$$f$$ is increasing in $$(a,b) \cup (c,d)$$

Option B is Correct

#### Find the intervals in which the function $$f(x) = 2\,x^3 -15\,x^2 + 36\,x+1$$ is increasing or decreasing?

A $$f$$ is increasing  in $$(-\infty,2) \cup(3,\infty)$$ and decreasing in $$(2,3)$$

B $$f$$ is increasing  in $$(-\infty,1) \cup(4,\infty)$$  and decreasing in $$(1,4)$$

C $$f$$ is increasing  in $$(-\infty,5) \cup(6,\infty)$$ and decreasing in $$(5,6)$$

D $$f$$ is increasing  in $$(-\infty,7) \cup(9,\infty)$$  and decreasing in $$(7,9)$$

×

For interval of increase $$\to$$$$f' (x) >0$$

For interval of decrease $$\to$$$$f' (x) <0$$

$$\Rightarrow 6\,x^2-30\,x+36>0$$

$$= 6(x^2-5\,x+6)>0$$

$$=6(x-2)(x-3)>0$$

$$=(x-2)(x-3)>0$$

 Interval $$(x-3)$$ $$(x-2)$$ $$f'$$ $$f$$ $$x>3$$ $$+$$ $$+$$ $$+$$ Increasing $$2 \(f$$ is increasing in $$(-\infty,2)\cup(3,\infty)$$

$$f$$ is decreasing in $$(2,3)$$

### Find the intervals in which the function $$f(x) = 2\,x^3 -15\,x^2 + 36\,x+1$$ is increasing or decreasing?

A

$$f$$ is increasing  in $$(-\infty,2) \cup(3,\infty)$$ and decreasing in $$(2,3)$$

.

B

$$f$$ is increasing  in $$(-\infty,1) \cup(4,\infty)$$  and decreasing in $$(1,4)$$

C

$$f$$ is increasing  in $$(-\infty,5) \cup(6,\infty)$$ and decreasing in $$(5,6)$$

D

$$f$$ is increasing  in $$(-\infty,7) \cup(9,\infty)$$  and decreasing in $$(7,9)$$

Option A is Correct

#### Find the intervals in which the function $$f(x) = \,x -2\,sin\,x$$ is increasing or decreasing $$(0\leq x\leq 2\,\pi)$$

A $$f$$ is increasing  in $$\left(\dfrac{\pi}{3},\dfrac{5\pi}{3}\right)$$ and decreasing in $$\left(0,\dfrac{\pi}{3}\right) \cup \left(\dfrac{5\pi}{3},2\,\pi\right)$$

B $$f$$ is increasing  in $$\left(0,\pi\right) \cup\left(\dfrac{5\,\pi}{3},2\,\pi\right)$$ and decreasing in $$\left(0,\dfrac{\pi}{3}\right) \cup \left(\pi,\dfrac{5\pi}{3}\right)$$

C $$f$$ is increasing  in $$\left(\dfrac {\pi}{2},\pi\right)$$ and decreasing in $$\left(0,\dfrac{\pi}{2}\right) \cup \left(\pi,2\,\pi\right)$$

D $$f$$ is increasing  in $$\left(\dfrac{\pi}{6},\dfrac{5\pi}{6}\right)$$ and decreasing in $$\left(0,\dfrac{\pi}{6}\right) \cup \left(\dfrac{5\pi}{6},2\,\pi\right)$$

×

For interval of increase $$\to$$$$f' (x) >0$$

For interval of decrease $$\to$$$$f' (x) <0$$

$$f'(x) = 1-2\,cos\,x$$

$$f'(x) > 0 \Rightarrow 1-2\,cos\,x > 0\,$$

$$\Rightarrow cos\,x<\dfrac{1}{2}$$

$$cos\,x= \dfrac{1}{2}$$  ,  when  $$x= \dfrac{\pi}{3},\dfrac{5\pi}{3}$$ Now Interval $$f'(x)= 1-2\,cos\,x$$ $$f(x)= x-2\,sin\,x$$ $$0 \(f$$ is increasing in $$\left(\dfrac{\pi}{3},\dfrac{5\,\pi}{3}\right)$$ $$f$$ is decreasing in $$\left(0,\dfrac{\pi}{3}\right) \cup \left(\dfrac{5\pi}{3},2\,\pi\right)$$ ### Find the intervals in which the function $$f(x) = \,x -2\,sin\,x$$ is increasing or decreasing $$(0\leq x\leq 2\,\pi)$$

A

$$f$$ is increasing  in $$\left(\dfrac{\pi}{3},\dfrac{5\pi}{3}\right)$$

and decreasing in $$\left(0,\dfrac{\pi}{3}\right) \cup \left(\dfrac{5\pi}{3},2\,\pi\right)$$

.

B

$$f$$ is increasing  in $$\left(0,\pi\right) \cup\left(\dfrac{5\,\pi}{3},2\,\pi\right)$$

and decreasing in $$\left(0,\dfrac{\pi}{3}\right) \cup \left(\pi,\dfrac{5\pi}{3}\right)$$

C

$$f$$ is increasing  in $$\left(\dfrac {\pi}{2},\pi\right)$$

and decreasing in $$\left(0,\dfrac{\pi}{2}\right) \cup \left(\pi,2\,\pi\right)$$

D

$$f$$ is increasing  in $$\left(\dfrac{\pi}{6},\dfrac{5\pi}{6}\right)$$

and decreasing in $$\left(0,\dfrac{\pi}{6}\right) \cup \left(\dfrac{5\pi}{6},2\,\pi\right)$$

Option A is Correct      #### Observe the following graph and indicate the open interval of decrease of function $$'f'$$.

A $$f$$  is decreasing in $$(a,b) \cup (c,d)$$

B $$f$$ is decreasing in $$(b,c) \cup (d,e)$$

C $$f$$ is decreasing in $$(a,c) \cup (d,e)$$

D $$f$$ is decreasing in $$(b,d) \cup (d,e)$$

×

Observe that $$f$$ is decreasing (falling graph) in (a,b) , then increasing (rising graph) in (b,c) , decreasing  in (c,d) and increasing in (d,e) again.

$$\therefore$$ Interval of increase is $$(b,c)\cup(d,e)$$

Interval of decrease is $$(a,b)\cup(c,d)$$

$$\therefore$$ Correct option is (A).

### Observe the following graph and indicate the open interval of decrease of function $$'f'$$. A

$$f$$  is decreasing in $$(a,b) \cup (c,d)$$

.

B

$$f$$ is decreasing in $$(b,c) \cup (d,e)$$

C

$$f$$ is decreasing in $$(a,c) \cup (d,e)$$

D

$$f$$ is decreasing in $$(b,d) \cup (d,e)$$

Option A is Correct

# Increasing Test (Ist Derivative Test)

• Consider the graph of an increasing function as shown.  • Now, we draw tangents to the same curve at some arbitrary points, say P and Q.  The slope of the tangent at point $$P=\tan\theta_2$$

The slope of the tangent at point $$Q=\tan\theta_1$$

Here, $$\theta_1$$ and $$\theta_2$$ are acute angles.

Thus, the slopes, $$\tan\theta_1$$ and $$\tan\theta_2$$ are positive.

• So, we can say that an increasing function in an interval has positive slope.
• So, for an increasing function,

$$f'(x)>0$$

## Finding interval of increase of function

• Let the function be $$f(x)=2x^2+x+2$$.
• We will find interval for the increase of this function.
• We know for an increasing function,

$$f'(x)>0$$

Step 1: Find $$f'(x)$$

$$f'(x)=\dfrac{d}{dx}(2x^2+x+2)$$

$$f'(x)=4x+1$$

Step 2: Apply condition of an increasing function.

$$f'(x)>0$$

$$4x+1>0$$

$$4x>-1$$

$$x>\dfrac{-1}{4}$$

So, the interval for the increase of the function is

$$\left(\dfrac{-1}{4},\;\infty\right)$$

#### Find the interval in which the function $$f(x) = 10 -6x-2x^2$$ is increasing.

A $$\left(\dfrac{3}{2}, \infty\right)$$

B $$\left(-\infty,\dfrac{5}{4} \right)$$

C $$\left(\dfrac{5}{4},\infty \right)$$

D $$\left(-\infty,-\dfrac{3}{2} \right)$$

×

For interval of increase $$\to$$ $$f'(x) > 0$$

$$\Rightarrow -6 -4x> 0$$

$$= x< - \dfrac{6}{4}$$

$$= x< - \dfrac{3}{2}$$

$$\Rightarrow x\,\varepsilon \left(-\infty, -\dfrac{3}{2}\right)$$

### Find the interval in which the function $$f(x) = 10 -6x-2x^2$$ is increasing.

A

$$\left(\dfrac{3}{2}, \infty\right)$$

.

B

$$\left(-\infty,\dfrac{5}{4} \right)$$

C

$$\left(\dfrac{5}{4},\infty \right)$$

D

$$\left(-\infty,-\dfrac{3}{2} \right)$$

Option D is Correct    #### Find the interval in which the function $$f(x) = 3 - 5\,x - 2\,x^2$$ is decreasing?

A $$\left(-\infty,\dfrac{1}{2} \right)$$

B $$\left(-\dfrac{5}{4},\infty \right)$$

C $$\left(\dfrac{5}{4},\infty \right)$$

D $$\left(\dfrac{3}{2},\infty \right)$$

×

For interval of decrease $$\to$$ $$f' (x) <0$$

$$\Rightarrow -5 -4x<0$$

$$=4x>-5$$

$$= x > -\dfrac{5}{4}$$

$$= x\,\varepsilon \left(-\dfrac{5}{4}, \infty\right)$$

### Find the interval in which the function $$f(x) = 3 - 5\,x - 2\,x^2$$ is decreasing?

A

$$\left(-\infty,\dfrac{1}{2} \right)$$

.

B

$$\left(-\dfrac{5}{4},\infty \right)$$

C

$$\left(\dfrac{5}{4},\infty \right)$$

D

$$\left(\dfrac{3}{2},\infty \right)$$

Option B is Correct