Find the f intervals for the following two functions. f(x)=(x-2)^2(x+1)^2, f(x)=x+4/x^2.

Almintas2l 2022-07-23 Answered
Finding f intervals
How would I find the f intervals for the following two functions.
f ( x ) = ( x 2 ) 2 ( x + 1 ) 2
using the chain rule I got ( x 2 ) 2 ( 2 ) ( x + 1 ) ( 1 ) + ( 2 ) ( x 2 ) ( 1 ) ( x + 1 ) 2
then I got f decrease ( , 1 ]
and f increase [ 2 , )
but the area between -1 and 2 in confusing me.
My second function is f ( x ) = x + 4 x 2
differentiating I got x 3 8 x 3
so I got f increase ( , 0 ) and [ 2 , )
f decrease (0, 2]
but did I do this correctly.
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Answers (1)

Bianca Chung
Answered 2022-07-24 Author has 16 answers
Step 1
As you did f′(x) correctly, we have f ( x ) = 2 ( x 2 ) ( x + 1 ) ( 2 x 1 )
Step 2
When:
- x 1    , ( x 2 ) 0 and then two terms ( x + 1 ) and ( 2 x 1 ) are negative. So f′(x) is negative.
- x 1 2    , ( x 2 ) 0 and ( x + 1 ) > 0 and ( 2 x 1 ) < 0 is negative, so f′(x) is positive.
- x 2    , ( x 2 ) 0 and then two terms ( x + 1 ) and ( 2 x 1 ) are positive, so f′(x) is nagative again.
- And if x > 2 all terms are positive and so f′(x) is positive again.
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