Calculus, solving for increasing/decreasing and concavityDetermine the intervals where the function is increasing and where it is decreasing. (Enter your answers using interval notation.) f ( x ) = ln ⁡ ( x ) x Find intervals of concavity for the graph of the function f ( x ) = ln ⁡ ( x ) x .I have already found the first and second derivative but I am confused on how to solve for it f ′ ( x ) = − ln ⁡ ( x ) − 1 x 2 f ″ ( x ) = 2 ln ⁡ ( x ) − 3 x 3

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Calculus, solving for increasing/decreasing and concavityDetermine the intervals where the function is increasing and where it is decreasing. (Enter your answers using interval notation.)  f  (  x  )  =            ln      ⁡      (      x      )        x  Find intervals of concavity for the graph of the function   f  (  x  )  =            ln      ⁡      (      x      )        x  .I have already found the first and second derivative but I am confused on how to solve for it       f    ′    (  x  )  =  −            ln      ⁡      (      x      )      −      1              x      2            f    ″    (  x  )  =            2      ln      ⁡      (      x      )      −      3              x      3

Haylee Branch · Accepted Answer

Step 1So let&#039;s first begin with finding out the intervals where the function is increasing and decreasing.To do this, we would find the critical numbers of the first derivative. (That is, what values of x, set       f    ′    (  x  )  =  0? Solving this, we get:  x  =  e     and     x  =  0Now that you have your critical numbers, you can form your intervals. This would be: (negative infinity to 0, 0 to e and e to infinity)You do the same thing for the second derivative to find intervals for concavity

mocatgesex · Accepted Answer

Step 1You have got       f    ′    (  x  )  =                    1        −        ln        ⁡        x                    x        2              ..If you solve   1  −  ln  ⁡  x  =  0 you get   x  =  e  .. Note that       x    2   is positive on   (  0  ,  ∞  ) which is the domain of f. So, you need to study the sign of f on (0,e) and   (  e  ,  ∞  )  .. You get                           (        0        ,        e        )                    (        e        ,        ∞        )                                      s          i          g          n                (                  f          ′                )                            +                    −            Step 2Thus, f is increasing on (0,e) and decreasing on   (  e  ,  ∞  )  ..Proceed in a similar way with the second derivative. (Note that       x    3   is positive on   (  0  ,  ∞  ) and so you only need to study the sign of the numerator.)

Determine the intervals where the function is increasing and where it is decreasing. (Enter your answers using interval notation.) f(x)=ln(x)/x.

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