Classify the critical point $x=2$ of $f(x)=(x-2)^{17}\left(x+5\right)^{24}$ Question 


  Let $f(x)=(x-2)^{17}(x+5)^{24}$. Then is $x=2$ a maximum, minimum, or neither?


Book's Approach 


  $f'''\left(2\right)\ne 0$. S
  Since the odd integral derivative of the function is non-zero, $x=2$ is neither a minimum nor maximum


Please Explain 

How did they conclude odd integral derivative
of the function is non-zero?

$f'''(2)\ne 0$ it is $2^{nd}$derivative. I don't how they did it.
The hint:

For $x&gt;2$ we have $f(x)&gt;0$ and for $x&lt;2$ around $2$ we have $f(x)&lt;0$.
Just a typo in the book. Should say: $f''(2)=0$, therefore neither maximum, nor minimum.

   f"(x) &lt; 0 - local maximum
   f"(x) &gt; 0 - local minimum
   f"(x) = 0 (your case)- neither of those