IXL - Find intervals of increase and decrease (Calculus practice)

Determine where the function

(

)

is increasing and where it is decreasing.

(

)

–

Express each answer as a list of disjoint, open intervals. Write one interval in each box. You can add or remove boxes. Remove all the boxes to indicate that the function is never increasing or never decreasing.

Increasing:

Decreasing:

ref_doc_title.

Learn with an example

question

Determine where the function

(

)

is increasing and where it is decreasing.

(

)

–

Increasing:

Decreasing:

key idea

′

(

)

for every number

in an open interval, then the function

is increasing on that interval.

′

(

)

for every number

in an open interval, then the function

is decreasing on that interval.

solution

To find where

(

)

is increasing or decreasing, determine where

′

(

)

and where

′

(

)

Start by calculating

′

(

)

(

)

–

′

(

)

–

Now, find where

′

(

)

is zero or does not exist. In other words, identify the critical points of

Since

′

(

)

exists for every value of

solve the equation

′

(

)

′

(

)

–

(

–

)

(

)

(

–

)

–

or x = 0 or x

These critical points split the domain of

into four intervals:

(

–

∞

–

)

(

–

)

(

)

and

(

∞

)

Use a sign chart to determine whether

′

is positive or negative on each interval. Since the derivative factors nicely as

′

(

)

(

)

(

–

)

the sign of

′

(

)

is the product of the sign of each factor.

	12	x	x + 2	x – 4	f ′ ( x )
( – ∞ , – 2 )	+	—	—	—	—
( – 2 , 0 )	+	—	+	—	+
( 0 , 4 )	+	+	+	—	—
( 4 , ∞ )	+	+	+	+	+

So,

′

is positive on

(

–

)

and

(

∞

)

and negative on

(

–

∞

–

)

and

(

)

That means the function

(

)

–

is increasing on

(

–

)

and

(

∞

)

and decreasing on

(

–

∞

–

)

and

(

)

You can see this in the graph of the function.

$Confetti$

Excellent!

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M.2Find critical points of functions

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