What is the critical point of this function?The problem reads: f ( x ) = 7 e 2 x x + 4..I am unsure of how to approach this problem to find the derivative. If someone could break down the steps that would be greatly appreciated.Also, the question asks for intervals of the increasing and decreasing parts of the function. How would I figure this out? I'm thinking I'd use a sign chart. But if you have any other useful methods, I am all ears, or rather eyes.

Question

What is the critical point of this function?The problem reads:   f  (  x  )  =  7            e              2        x              x    +  4..I am unsure of how to approach this problem to find the derivative. If someone could break down the steps that would be greatly appreciated.Also, the question asks for intervals of the increasing and decreasing parts of the function. How would I figure this out? I&#039;m thinking I&#039;d use a sign chart. But if you have any other useful methods, I am all ears, or rather eyes.

Cynthia Lester · Accepted Answer

Step 1Let   f  (  x  )  =  7                    e                  2          x                    x        +  4 for   x  ≠  0. Then, we have for   x  ≠  0.                    (1)                              f          ′                (        x        )        =        7                  e                      2            x                                    (                                    2              x                                −                                    1                              x                2                                              )                    where we used the product rule   (  g  h      )    ′    =      g    ′    h  +  g      h    ′   with   g  (  x  )  =      e          2      x      ,   h  (  x  )  =  1      /    x,       g    ′    (  x  )  =  2      e          2      x      , and       h    ′    (  x  )  =  −  1      /        x    2  .Step 2Now, we note from (1) that       f    ′    (  x  )  =  0 when   x  =  1      /    2.Similarly, from (1), we see that when   x  &amp;gt;  1      /    2,       f    ′    &amp;gt;  0 and f is increasing. When   0  &amp;lt;  x  &amp;lt;  1      /    2 or   x  &amp;lt;  0,       f    ′    &amp;lt;  0 and f is decreasing.

The problem reads: f(x)=7(e^2x)/x+4.

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