FINDING INCREASING OR DECREASING INTERVALS

Procedure to find where the function is increasing or decreasing :

  • Find the first derivative. 
  • Then set f'(x)  =  0
  • Put solutions on the number line.
  • Separate the intervals.
  • Choose random value from the interval and check them in the first derivative.
  • If f(x) > 0, then the function is increasing in that particular interval.
  • If f(x) < 0, then the function is decreasing in that particular interval.

Example 1 :

Find the intervals in which

f(x) = 2x³+x²-20x

is increasing or decreasing

Solution :

f(x)  =  2x3 + x2 - 20x

Step 1 :

f'(x)  =  6x² + 2x - 20

÷ by 2 ⇒ 3x²+x-10

Step 2 :

  f'(x)  =  0

3x2+x-10  =  0

(3x - 5) (x + 2)  =  0    

3x-5  =  0

3x  =  5

x  =  5/3

x+2  =  0

x  =  -2

Step 3 :

We can split this into three intervals (-∞,-2) (-2,5/3) (5/3,∞).

Step 4 :

Now let us see the given function is increasing or decreasing in which intervals.

Interval

-∞ < x < -2

-2 < x < 5/3

5/3 < x < ∞

3x-5

-

+

+

x-2

-

-

+

f'(x)

+

-

+

Step 5 :

The given is increasing on (-∞,-2] ∪ [5/3,-∞) and decreasing on [-2,5/3]

Example 2 :

Find the intervals in which

f(x) = x³ - 3 x + 1

is increasing or decreasing

Solution :

f(x)  =  x³ - 3 x + 1

f'(x)  =  3x² - 3

÷ by 3 ⇒ x² - 1

f'(x)  =  0

x² - 1  =  0

(x + 1) (x - 1)  =  0

x+1  =  0

x  =  -1

x-1  =  0

x  =  1

We can split this as three intervals (-∞,-1) (-1,1) (1,∞).

Now let us see the given function is increasing or decreasing in which intervals.

Interval

-∞ < x < -1

-1 < x < 1

1 < x < ∞

x+1

-

+

+

x-1

-

-

+

f'(x)

+

-

+

The given is increasing on (-∞,-1] ∪ [1,∞) and decreasing on [-1, 1].

Example 3 :

Find the intervals in which f (x)  =  x - 2 sin x is increasing or decreasing

Solution :

f(x)  =  x - 2 sin x

f'(x)  =  1 - 2 cos x

f'(x)  =  0

1 - 2cos x  =  0

-2 cos x  =  -1

cos x  =  1/2

x  =  cos ⁻¹(1/2)

  x  =  Π/3, 5Π/3

We can split this as three intervals (0,Π/3) (Π/3,5Π/3) (5Π/3,2Π).

Now let us see the given function is increasing or decreasing in which intervals.

Interval

0 < x < Π/3

Π/3 < x < 5Π/3

5Π/3 < x < 2Π

1 - 2cos x

-

+

-

f'(x)

-

+

+

The given is increasing on [Π/3, 5Π/3] and decreasing on (0,Π/3] ∪ [5Π/3,2Π).

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