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Question
If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\] then `f' (x)` is equal to ____________ .
Options
\[- 2x + 9\text { for all } x \in R\]
\[2x - 9 \text { if }4 < x < 5\]
\[- 2x + 9, \text { if }4 < x < 5\]
none of these
Solution
\[- 2x + 9 \text{ for }4 < x < 5 \]
\[\text { We have,} f\left( x \right) = \left| x^2 - 9x + 20 \right| \]
\[f\left( x \right) = \begin{Bmatrix}x^2 - 9x + 20, - \infty < x \leq 4 \\ - \left( x^2 - 9x + 20 \right), 4 < x < 5 \\ x^2 - 9x + 20, 5 \leq x < \infty\end{Bmatrix}\]
\[ \Rightarrow f\left( x \right) = \begin{Bmatrix}2x - 9, - \infty < x \leq 4 \\ - 2x + 9, 4 < x < 5 \\ 2x - 9, 5 \leq x < \infty\end{Bmatrix}\]
\[ \therefore f'\left( x \right) = - 2x + 9 \text { for } 4 < x < 5 \]
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