## 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 increasing 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  =  03x  =  5x  =  5/3 x+2  =  0x  =  -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 < 5/35/3 < 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  =  0x  =  -1 x-1  =  0x  =  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 < 11 < 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.

 Interval0 < x < Π/3Π/3 < x < 5Π/35Π/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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