FINDING INCREASING OR DECREASING INTERVALS

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    

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.

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

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.

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.

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

The following graphs show the derivative of 𝒇, 𝒇. Identify the intervals when 𝒇 is increasing and decreasing. Include a justification statement

Example 4 :.

Solution :

• In the graph of first derivative, the part which lies below the x-axis will have negative slope. Then it is decreasing function in that particular interval.
• In the graph of first derivative, the part which lies above the x-axis will have positive slope. Then it is increasing function in that particular interval.

By observing the graph given above, (-∞, -1.5) and (-0.5, ∞) the curve is above the x-axis. Then it is increasing function in the interval mentioned.

By observing the graph above, (-1.5, -0.5) the curve is below the x-axis. Then it is decreasing function in the interval mentioned.

Example 5 :

Solution :

By observing the graph given above, (-1, 0) and (3, ∞) the curve is above the x-axis. Then it is increasing function in the interval mentioned.

By observing the graph above, (-∞, -1) and (0, 3) the curve is below the x-axis. Then it is decreasing function in the interval mentioned.

For each function, find the intervals where it is increasing and decreasing, and JUSTIFY your conclusion. Construct a sign chart to help you organize the information,

Example 6 :

f(x) = x³ - 12x + 1

Solution :

f(x) = x³ - 12x + 1

f'(x) = 3x² - 12(1) + 0

=  3x² - 12

f'(x) = 0

3x² - 12 = 0

3(x² - 4) = 0

3 (x + 2) (x - 2) = 0

x = -2 and x = 2

Increasing intervals are (-∞, -2) U (2, ∞) and decreasing interval is (-2, 2).

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