Let $f (x) = | | x - 6 | - | x - 8 | | - 3 | x | .$ Then the number of local extremum point(s) is

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Solution

Given: $f (x) = | | x - 6 | - | x - 8 | | - 3 | x |$
$\Rightarrow f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 3 x + 2, & x < 0 2 - 3 x, & 0 \leq x < 6 14 - 5 x, & 6 \leq x < 7 - x - 14, & 7 \leq x < 8 2 - 3 x, & x \geq 8 \end{matrix}$

Figure of $f (x) :$

Clearly from the diagram only $x = 0$ is the point of local extremum.

Hence, the number of local extremum point is $1.$

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