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First Principle of Differentiation

If fx = |x2 −...

Question

If $f (x) = | x^{2} - 5 x + 6 |$ , then $f^{'} (x)$ equals

A

2 x - 5

for

2 < x < 3

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B

5 - 2 x

for

2 < x < 3

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C

2 x - 5

for

x < 2

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D

5 - 2 x

for

x < 3

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Solution

The correct option is C $2 x - 5$ for $x < 2$
We know that,
$| x^{2} - 5 x + 6 | = | (x - 3) (x - 2) |$
So the function will be,
$f (x) = {\begin{matrix} x^{2} - 5 x + 6 if x \geq 3 or x \leq 2 - (x^{2} - 5 x + 6) if 2 < x < 3 \end{matrix} \Rightarrow f^{'} (x) = {\begin{matrix} 2 x - 5 if x > 3 or x < 2 - 2 x + 5 if 2 < x < 3 \end{matrix}$

$f (x)$ is non-differentiable at $x = 2 & x = 3$

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