The number of points of local maxima and loca | vedclass

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(D) Given the function $f(x) = |x^2 - 2|x||$. Since $x^2 = |x|^2$,we can write $f(x) = ||x|^2 - 2|x||$.Let $t = |x|$,where $t \ge 0$. Then $g(t) = |t^

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$7$

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(D) Given the function $f(x) = |x^2 - 2|x||$. Since $x^2 = |x|^2$,we can write $f(x) = ||x|^2 - 2|x||$.Let $t = |x|$,where $t \ge 0$. Then $g(t) = |t^2 - 2t|$.The graph of $g(t)$ for $t \ge 0$ has a local maximum at $t=1$ where $g(1) = |1-2| = 1$,and local minima at $t=0$ where $g(0)=0$ and $t=2$ where $g(2)=0$.Since $t = |x|$,the points of local maxima for $f(x)$ occur at $x = \pm 1$. Thus,$M = 2$.The points of local minima for $f(x)$ occur at $x = 0, 2, -2$. Thus,$m = 3$.Therefore,the value of $2M + m = 2(2) + 3 = 4 + 3 = 7$.

The number of points of local maxima and local minima of the function $f(x) = |x^2 - 2|x||$ in $\mathbb{R}$ are $M$ and $m$ respectively. Then,the value of $2M + m$ is -

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