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

Question

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.

Answer 1

Clearly $f(2)=0$ and just check the values of $f$ near $2$. If $x<2$ then $f(x) <0$ and if $x>2$ then $f(x) >0$. Thus the point $x=2$ is neither a minimum nor a maximum of $f$. We don't need derivatives for such simple problems.

Answer 2

The hint:

For $x>2$ we have $f(x)>0$ and for $x<2$ around $2$ we have $f(x)<0$.

Answer 3

Just a typo in the book. Should say: $f''(2)=0$, therefore neither maximum, nor minimum.

   f"(x) < 0 - local maximum
   f"(x) > 0 - local minimum
   f"(x) = 0 (your case)- neither of those