Let $f(x)=(x-2)^{17}(x+5)^{24}$. Then

• A

  $f$ does not have a critical point at $x=2$
• B

  $f$ has a minimum at $x=2$
• Correct answer
  C

  $f$ has neither a maximum nor a minimum at $x=2$
• D

  f has a maximum at $x=2$

Solution

$\begin{aligned} & f(x)=(x-2)^{17}(x+5)^{24} \\ \Rightarrow & f^{\prime}(x)=17(x-2)^{16}(x+5)^{24}+24(x-2)^{17}(x+5)^{23}=(x-2)^{16}(x+5)^{23}(17 x+85+24 x-48)=(x-2)^{16}(x+5)^{23}(41 x+37) \end{aligned}$

Hence, the correct answer is option: 'C'.