Question

If $f\left(x\right)=|x^2-5x+6|$, then which among the following is/are CORRECT?

• A. $f^{\prime }\left(x\right)=2x-5,\text{if}x>3\text{or}x<2$
• B. $f^{\prime }\left(x\right)=2x-5$ for all $x$
• C. $f^{\prime }\left(x\right)=5-2x,\text{if}2<x<3$
• D. $f^{\prime }\left(x\right)=2x-5,\text{if}x\ge 3\text{or}x\le 2$

Answer 1

Answer: (A), (C)

$f^{\prime }\left(x\right)=5-2x,\text{if}2<x<3$

We have,
$f\left(x\right)=|x^2-5x+6|$

$\Rightarrow f\left(x\right)=\{\begin{array}{cc}{x}^2-5x+6,& \text{if}x\ge 3\text{or}x\le 2\\ -(x^2-5x+6),& \text{if}2<x<3\end{array}$

$\Rightarrow f^{\prime }\left(x\right)=\{\begin{array}{cc}2x-5,& \text{if}x>3\text{or}x<2\\ 5-2x,& \text{if}2<x<3\end{array}$

$\because$ At $x=2,3$
$f\left(x\right)$ is not differentiable