Let f be a continuous,periodic even function | vedclass

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(A) Given that $f(x)$ is a continuous,periodic even function with period $T = 4$. Since $f(0) = 1$ and $f(2) = -1$,by the Intermediate Value Theorem,t

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$10$

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(A) Given that $f(x)$ is a continuous,periodic even function with period $T = 4$. Since $f(0) = 1$ and $f(2) = -1$,by the Intermediate Value Theorem,there must be at least one root in the interval $(0, 2)$. Because $f$ is an even function,$f(-x) = f(x)$,so there must be at least one root in the interval $(-2, 0)$. Thus,there are at least $2$ roots in one period interval $(-2, 2)$. The interval $[-10, 10]$ has a length of $20$,which covers $20/4 = 5$ periods. Therefore,the minimum number of roots in $[-10, 10]$ is $2 	imes 5 = 10$.

Let $f$ be a continuous,periodic even function defined on $\mathbb{R}$ such that $f(0) = 1$,$f(2) = -1$ and the period of $f$ is $4$. The minimum number of roots of the equation $f(x) = 0$ in the interval $[-10, 10]$ will be:

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