Can anyone confirm if my answer is right?Find the critical points and the intervals of the increase and decrease of the function f ( x ) = ( x + 5 ) 2 ( x − 2 ) 5 The critical points are: − 3 , − 5 , 2The four intervals -from left to right these open intervals are ( − ∞ , − 5 ) , ( − 5 , − 3 ) , ( − 3 , 2 ) and ( 2 , ∞ )This part I am not sure of:On the interval ( − ∞ , − 5 ), f(x) is decreasingOn the interval ( − 5 , − 3 ), f(x) is decreasingOn the interval ( − 3 , 2 ), f(x) is increasingOn the interval ( 2 , ∞ ), f(x) is increasing

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Can anyone confirm if my answer is right?Find the critical points and the intervals of the increase and decrease of the function   f  (  x  )  =  (  x  +  5      )    2    (  x  −  2      )    5  The critical points are:   −  3  ,     −  5  ,     2The four intervals -from left to right these open intervals are   (  −  ∞  ,  −  5  )  ,     (  −  5  ,  −  3  )  ,     (  −  3  ,  2  )     and     (  2  ,  ∞  )This part I am not sure of:On the interval   (  −  ∞  ,  −  5  ), f(x) is decreasingOn the interval   (  −  5  ,  −  3  ), f(x) is decreasingOn the interval   (  −  3  ,  2  ), f(x) is increasingOn the interval   (  2  ,  ∞  ), f(x) is increasing

Rayan Ali · Accepted Answer

Step 1  f      ´    (  x  )  =  [  (  x  +  5      )    2    ]      ´    (  x  −  2      )    5    +  (  x  +  5      )    2    [  (  x  −  2      )    5    ]      ´    f      ´    (  x  )  =  2  (  x  +  5  )  (  x  −  2      )    5    +  5  (  x  +  5      )    2    (  x  −  2      )    4    f      ´    (  x  )  =  (  x  +  5  )  (  x  −  2      )    4    [  2  (  x  −  2  )  +  5  (  x  +  5  )  ]  f      ´    (  x  )  =  7  (  x  +  5  )  (  x  +  3  )  (  x  −  2      )    4  Step 2Your break points are:   −  5  ,  −  3  ,  2.Using a sign analysis you find out the intervals to be:Increasing on:   (  −      ∞    ,  −  5  )  ∪  (  −  3  ,  2  )  ∪  (  2  ,      ∞    ).Decreasing on:   (  −  5  ,  −  3  )

dammeym · Accepted Answer

Step 1First your critical points should be   x  =  −  5  ,  −  1.5  ,  2Since       f    ′    (  x  )  =  4      x    3    +  18      x    2    −  22  x  −  60 and we look at the interval   ∞  &amp;lt;  x  &amp;lt;  −  5 and its sign (f&#039;(x)) is negative in that interval so it is decreasing. Next we look at   −  5  &amp;lt;  x  &amp;lt;  −  1.5 where the sign of the differential is positive so it is increasing.Step 2Then   −  1.5  &amp;lt;  x  &amp;lt;  2 where again the differential is negative so decreasing! Finally   2  &amp;lt;  x  &amp;lt;  ∞ where the sign is positive so increasing!

Find the critical points and the intervals of the increase and decrease of the function f(x)=(x+5)^2(x-2)^5.

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