Find the Intervals of Increase or Decrease, the concavity, and point of inflection for: f ( x ) = ( 1 − x ) e − x . f ( x ) = ( 1 − x ) e − x = 1 − x e x Quotient Rule: f ′ ( x ) = e x ⋅ [ 1 − x ] ′ − ( 1 − x ) ⋅ [ e x ] ′ = e x ( − 2 + x ) What do I do now? Take the derivative again?

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Find the Intervals of Increase or Decrease, the concavity, and point of inflection for:   f  (  x  )  =  (  1  −  x  )      e          −      x      .                    f        (        x        )                            =        (        1        −        x        )                  e                      −            x                                                            =                              1            −            x                                e                          x                                          Quotient Rule:                              f          ′                (        x        )                            =                  e          x                ⋅        [        1        −        x                  ]          ′                −        (        1        −        x        )        ⋅        [                  e          x                          ]          ′                                                  =                  e          x                (        −        2        +        x        )            What do I do now? Take the derivative again?

Kyle George · Accepted Answer

Step 1This will only be a hint:An interval of decrease means the function is decreasing. What does that mean about the derivative? Can you solve for x when       f    ′    (  x  )  &amp;lt;  0?An interval of increase means the function is increasing. What does that mean about the derivative? Can you solve for x when       f    ′    (  x  )  &amp;gt;  0?The concavity of the function at a point means that the change of the function at that point is either increasing or decreasing. This is therefore related to the 2nd derivative. Can you find when       f    ″    (  x  )  &amp;gt;  0? How about when       f    ″    (  x  )  &amp;lt;  0?Step 2A point of inflection is the point (or points) at which the 1st derivative is no longer changing. That is,       f    ″    (  x  )  =  0. Can you solve for x in this case as well?Notice how once you done all of these (admittedly tedious) calculations, you&#039;ll basically be able to plot the function even if you don&#039;t really know what the function actually looks like. Simply by considering the derivatives of the function and when they&#039;re 0, positive, or negative.

Find the Intervals of Increase or Decrease, the concavity, and point of inflection for: f(x)=(1-x)e^{-x}.

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