Question

If $f\left(x\right)=\left|x^2-9x+20\right|$, then f' (x) is equal to
(a) $-2x+9\mathrm{for}\mathrm{all}x\in R$
(b) $2x-9\mathrm{if}4<x<5$
(c) $-2x+9,\mathrm{if}4<x<5$
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{c}\right)-2x+9\mathrm{for}4<x<5 \\ \mathrm{We}\mathrm{have},f\left(x\right)=\left|x^2-9x+20\right| \\ f\left(x\right)=\left\{\begin{array}{c}{x}^2-9x+20,-\infty <x\le 4\\ -\left(x^2-9x+20\right),4<x<5\\ x^2-9x+20,5\le x<\infty \end{array}\right\} \\ \Rightarrow f\left(x\right)=\left\{\begin{array}{c}2x-9,-\infty <x\le 4\\ -2x+9,4<x<5\\ 2x-9,5\le x<\infty \end{array}\right\} \\ \therefore f\text{'}\left(x\right)=-2x+9\mathrm{for}4<x<5 \end{gathered}$$