The number of solutions of the equation |x^2 | vedclass

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(C) To find the number of solutions for the equation $|x^2 - 2|x|| = 2^x$,we analyze the intersection of the graphs of $f(x) = |x^2 - 2|x||$ and $g(x)

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$3$

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(C) To find the number of solutions for the equation $|x^2 - 2|x|| = 2^x$,we analyze the intersection of the graphs of $f(x) = |x^2 - 2|x||$ and $g(x) = 2^x$.$1$. The function $f(x) = |x^2 - 2|x||$ is an even function,meaning it is symmetric about the $y$-axis.$2$. For $x \ge 0$,$f(x) = |x^2 - 2x|$. The roots are at $x=0$ and $x=2$. The graph has a local maximum at $x=1$ with $f(1) = |1-2| = 1$.$3$. For $x $4$. The function $g(x) = 2^x$ is an increasing exponential function.$5$. By plotting both functions,we observe that the curve $g(x) = 2^x$ intersects the curve $f(x)$ at three distinct points: one in the interval $(-2, 0)$,one at $x=0$ (since $f(0)=0$ and $g(0)=1$,wait,let's re-evaluate: $f(0)=0, g(0)=1$,no intersection at $x=0$),and two for $x > 0$.$6$. Specifically,$g(x)$ intersects $f(x)$ at one point where $x  0$.$7$. Therefore,there are $3$ points of intersection.

The number of solutions of the equation $|x^2 - 2|x|| = 2^x$ is:

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