Determine the total number of critical points of the function f ( x ) = ( x + e x ) k , where k &gt; 0 is an integer

Question

Determine the total number of critical points of the function   f  (  x  )  =  (  x  +      e    x        )    k  , where   k  &amp;gt;  0 is an integer

Tolamaes04 · Accepted Answer

f    ′    (  x  )  =  k  (  x  +      e    x        )          k      −      1        ×  (  1  +      e    x    )  =  0has only one solution which is where   x  +      e    x    =  0 and that is the point that you want to approximate.The answer should be negative so   x  =  0.567 is problematic.

Zoagliaj · Accepted Answer

At critical points       f    ′    (  x  )  =  0 so  0  =  k  (  x  +      e    x        )          k      −      1        ×  (  1  +      e    x    )  k and   (  1  +      e    x    ) are positive so   0  =  x  +      e    x        x    n    =  −      e                  x                  n          +          1                     so   x  ≈  −  0.56714329

Determine the total number of critical points of the function f(x)=(x+e^x)^k, where k>0 is an integer

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