Question:
Choose the correct alternative in the following:
If $f(x)=\left|x^{2}-9 x+20\right|$, then $f^{\prime}(x)$ is equal to
A. $-2 x+9$ for all $x \in R$
B. $2 x-9$ if $4 C. $-2 x+9$ if $4 D. none of these
Solution:
$f(x)=\left|x^{2}-9 x+20\right|$
$=\left|x^{2}-4 x-5 x+20\right|$
$=|x(x-4)-5(x-4)|$
$f(x)=|(x-5)(x-4)|$
$\Rightarrow \mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}(\mathrm{x}-5)(\mathrm{x}-4), \mathrm{x} \geq 5 \text { and } \mathrm{x} \geq 4 \\ -(\mathrm{x}-5)(\mathrm{x}-4), 4<\mathrm{x}<5\end{array}\right.$
$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\left\{\begin{array}{c}(2 \mathrm{x}-9), \mathrm{x} \geq 5 \text { and } \mathrm{x} \geq 4 \\ -2 \mathrm{x}+9,4<\mathrm{x}<5\end{array}\right.$