Finding f intervalsHow 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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Finding f intervalsHow 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.

Bianca Chung · Accepted Answer

Step 1As you did f′(x) correctly, we have       f    ′    (  x  )  =  2  (  x  −  2  )  (  x  +  1  )  (  2  x  −  1  )Step 2When:-   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  )  &amp;gt;  0 and   (  2  x  −  1  )  &amp;lt;  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  &amp;gt;  2 all terms are positive and so f′(x) is positive again.

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

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