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.

Illustration Questions

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 \('f'\).

image
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

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

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

Illustration Questions

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

Illustration Questions

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

Illustration Questions

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 
\(2<x<3\) \(-\) \(+\) \(-\) Decreasing
\(x<2\) \(-\) \(-\) \(+\) Increasing 

\(f\) 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

Illustration Questions

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  \(x= \dfrac{\pi}{3},\dfrac{5\pi}{3} \)   

image

Now Interval \(f'(x)= 1-2\,cos\,x\) \(f(x)= x-2\,sin\,x\)
\(0<x<\dfrac{\pi}{3}\) \(-\) Decreasing 
\(\dfrac{\pi}{3}<x<\dfrac{5\pi}{3}\) \(+\) increasing
\(\dfrac{5\pi}{3}<x<2\,\pi\) \(-\) Decreasing 

image

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

image

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

image

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

Illustration Questions

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 \('f'\).

image
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

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