How to find where the function is decreasing/increasing/concave/convex f ( x ) = 2 1 + x 2 ?I need to find where this function is increasing, decreasing, concave and convex. I've found it's derivative: f ′ ( x ) = − 4 x ( 1 + x 2 ) 2 Now you're supposed to make either f ′ ( x ) &gt; 0 when it's increasing and f ′ ( x ) &lt; 0 when it's decreasing, but that gives:Increasing: x &lt; 0. Decreasing: x &gt; 0.But what does that actually mean? It's just confusing, usually when I solve these you get 2 solutions, so it's for example increasing on the interval of (-2,2). What does this one tell me? What's the easiest way to find where this function is increasing and decreasing?Then I also did the second derivative, which is: f ″ ( x ) = 4 ( 3 x 2 − 1 ) ( 1 + x 2 ) 3 How does this all help me find my solution?

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How to find where the function is decreasing/increasing/concave/convex   f  (  x  )  =            2              1        +                  x          2                    ?I need to find where this function is increasing, decreasing, concave and convex. I&#039;ve found it&#039;s derivative:      f    ′    (  x  )  =            −      4      x              (      1      +              x        2                    )        2            Now you&#039;re supposed to make either       f    ′    (  x  )  &amp;gt;  0 when it&#039;s increasing and       f    ′    (  x  )  &amp;lt;  0 when it&#039;s decreasing, but that gives:Increasing:   x  &amp;lt;  0. Decreasing:   x  &amp;gt;  0.But what does that actually mean? It&#039;s just confusing, usually when I solve these you get 2 solutions, so it&#039;s for example increasing on the interval of (-2,2). What does this one tell me? What&#039;s the easiest way to find where this function is increasing and decreasing?Then I also did the second derivative, which is:      f    ″    (  x  )  =            4      (      3              x        2            −      1      )              (      1      +              x        2                    )        3            How does this all help me find my solution?

Catherine Kirby · Accepted Answer

Step 11. The endpoints   ±  ∞ are always present. Your results say that f is increasing on   (  −  ∞  ,  0  ) and decreasing on   (  0  ,  ∞  ).Step 22. For convexity / concavity you need to do the same thing to f′′,       f    ″    &amp;gt;  0 implies convexity and       f    ″    &amp;lt;  0 implies concavity. Note that the numerator is the only interesing component of f′′ because the denominator is always   ≥  1, so it doesn&#039;t change the sign. Now to find the critical points, solve for   4  (  3      x    2    −  1  )  =  0  ⇔      x    2    −      1    3    =  0  ⇔  x  =  ±      1          3

How to find where the function is decreasing/increasing/concave/convex f(x)=2/(1+x^2)?

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