First derivative test to find where the function is increasing, and decreasingI have this function: f ( x ) = x 2 l o g ( x ) − 1 .and I want to find increasing intervals, and decreasing intervals. Here's what I did:- I've found domain in which the function is defined, and this is what I got: f : ] 0 , + ∞ [ ( w i t h o u t 1 ) − &gt; R.0 and 1 are critical points in which the function isn't defined (i.e if x = 0, the log is undefined, and if x = 1, the fraction is undefined).- I've computed first derivative, and this is what I got (I've used quotient rule): 2 x ∗ ( l o g ( x ) − 1 ) − x 2 ∗ 1 / x ( l o g ( x ) − 1 ) 2 I've rewritten it, as follows: 2 x ∗ ( l o g ( x ) − 1 ) − x ( l o g ( x ) − 1 ) 2 .the denominator is always positive so the sign depends only on the numerator. (I think the problem is in this step) on a line, I've chosen 4 different random numbers, 1/4, 1/2, 3/2, 2, and I've put each number in the first derivative, and this is what I got:- in the first interval (from 0, to 1/4), the original function is decreasing (minus sign)- same for the other intervalsI don't have the full solution of the exercise, therefore I don't knwo if my solution is correct, but I think is strange that in this function there aren't increasing intervals.- I think the problem is in the first derivative, but I'm sure I've used the quotient rule correctly.

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First derivative test to find where the function is increasing, and decreasingI have this function:   f  (  x  )  =            x      2              l      o      g      (      x      )      −      1      .and I want to find increasing intervals, and decreasing intervals. Here&#039;s what I did:- I&#039;ve found domain in which the function is defined, and this is what I got:   f  :  ]  0  ,  +  ∞  [  (  w  i  t  h  o  u  t     1  )  −  &amp;gt;  R.0  and 1 are critical points in which the function isn&#039;t defined (i.e if   x  =  0, the log is undefined, and if   x  =  1, the fraction is undefined).- I&#039;ve computed first derivative, and this is what I got (I&#039;ve used quotient rule):            2      x      ∗      (      l      o      g      (      x      )      −      1      )      −              x        2            ∗      1              /            x              (      l      o      g      (      x      )      −      1              )        2            I&#039;ve rewritten it, as follows:             2      x      ∗      (      l      o      g      (      x      )      −      1      )      −      x              (      l      o      g      (      x      )      −      1              )        2            .the denominator is always positive so the sign depends only on the numerator. (I think the problem is in this step) on a line, I&#039;ve chosen 4 different random numbers, 1/4, 1/2, 3/2, 2, and I&#039;ve put each number in the first derivative, and this is what I got:- in the first interval (from 0, to 1/4), the original function is decreasing (minus sign)- same for the other intervalsI don&#039;t have the full solution of the exercise, therefore I don&#039;t knwo if my solution is correct, but I think is strange that in this function there aren&#039;t increasing intervals.- I think the problem is in the first derivative, but I&#039;m sure I&#039;ve used the quotient rule correctly.

Tirioliwo · Accepted Answer

Step 1Assuming log stands for the natural logarithm, you can write the derivative as       f    ′    (  x  )  =            x      (      2      log      ⁡      (      x      )      −      3      )              (      log      ⁡      (      x      )      −      1              )        2              ..Step 2As   x  &amp;gt;  0, its sign depends only on the sign of the parenthesis in the numerator and we have that   x  &amp;lt;      e          3              /            2          ⟹        f    ′    (  x  )  &amp;lt;  0 so the function is decreasing in   ]  0  ,  1  [  ∪  ]  1  ,      e          3              /            2        [ and   x  &amp;gt;      e          3              /            2          ⟹        f    ′    (  x  )  &amp;gt;  0 so it&#039;s increasing in   ]      e          3              /            2        ,  +  ∞  [  .

I have this function: f(x)=x^2/(log(x)-1) and I want to find increasing intervals, and decreasing intervals.

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