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 can analyze the graphs of $f(x) = |x^2 - 2|x||$ and $g(x) = 2^x$.$1$. Ana

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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 can analyze the graphs of $f(x) = |x^2 - 2|x||$ and $g(x) = 2^x$.$1$. Analyze $f(x) = |x^2 - 2|x||$:- Since $f(x)$ is an even function $(f(x) = f(-x))$,the graph is symmetric about the $y$-axis.- For $x \ge 0$,$f(x) = |x^2 - 2x|$.- The roots of $x^2 - 2x = 0$ are $x = 0$ and $x = 2$.- Between $x = 0$ and $x = 2$,$x^2 - 2x$ is negative,so $|x^2 - 2x| = -(x^2 - 2x) = 2x - x^2$.- For $x > 2$,$x^2 - 2x$ is positive,so $|x^2 - 2x| = x^2 - 2x$.$2$. Analyze $g(x) = 2^x$:- This is an exponential growth function that is always positive and strictly increasing.$3$. Intersection points:- By plotting both functions,we observe the intersection points.- For $x - For $x > 0$,the function $f(x) = |x^2 - 2x|$ intersects $g(x) = 2^x$ at exactly one point (since $2^x$ grows much faster than $x^2 - 2x$ for large $x$).- Thus,there are a total of $3$ intersection points.Therefore,the number of solutions is $3$.

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

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