Increasing and decreasing intervals of a function f ( x ) = x 3 − 4 x 2 + 2, which of the following statements are true:(1) Increasing in ( − ∞ , 0 ), decreasing in ( 8 3 , + ∞ ).(2) Increasing in both ( − ∞ , 0 ), decreasing in ( 8 3 , + ∞ ).(3) decreasing in both ( − ∞ , 0 ), and ( 8 3 , + ∞ ).(4) Decreasing in ( − ∞ , 0 ), Increasing in ( 8 3 , + ∞ ).(5) None of the above. f ′ ( x ) = 0 = 3 x 2 − 8 x = 0 ⇒ x = 8 3 , x = 0 are the singular point/point of inflection.Could anyone tell me what next?

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Increasing and decreasing intervals of a function  f  (  x  )  =      x    3    −  4      x    2    +  2, which of the following statements are true:(1) Increasing in   (  −  ∞  ,  0  ), decreasing in   (      8    3    ,  +  ∞  ).(2) Increasing in both   (  −  ∞  ,  0  ), decreasing in   (      8    3    ,  +  ∞  ).(3) decreasing in both   (  −  ∞  ,  0  ), and   (      8    3    ,  +  ∞  ).(4) Decreasing in   (  −  ∞  ,  0  ), Increasing in   (      8    3    ,  +  ∞  ).(5) None of the above.      f    ′    (  x  )  =  0  =  3      x    2    −  8  x  =  0  ⇒  x  =      8    3    ,  x  =  0 are the singular point/point of inflection.Could anyone tell me what next?

Paige Paul · Accepted Answer

Step 1So, when you put       f    ′    (  x  )  =  0 and got   x  =  0  ,  8      /    3, it means that function takes the “u-turn” at those points. Now, we need to check what was happening before   x  =  0, what’s happening between 0 and 8/3 and what will happen after   x  =  8      /    3.We can take any two x’s such that   x  &amp;lt;  0 and we will find that if       x    1    &amp;lt;      x    2   then   f  (      x    1    )  &amp;lt;  f  (      x    2    ) or we can see that       f    ′    (  x  )  =  3      x    2    −  8  x is positive for any   x  &amp;lt;  0 and hence function is increasing in the interval   (  −  ∞  ,  0  ].Step 2And as we know that the function will take a “u-turn” at   x  =  0 so, the function will decrease between 0 and 8/3 and again after a “u-turn” at   x  =  8      /    3 it will increase. If you find this “u-turn” concept informal you can go for similar method above and, you will find that f is decreasing between   x  =  0 and   x  =  8      /    3, and finally it is increasing for   x  &amp;gt;  8      /    3

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