**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 = 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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