The graphs of the polynomial x^{2}-1 and cos | vedclass

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(A) Let $f(x) = x^{2}-1$ and $g(x) = \cos x$. We are looking for the number of solutions to the equation $x^{2}-1 = \cos x$. Observe the behavior of b

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at exactly two points

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(A) Let $f(x) = x^{2}-1$ and $g(x) = \cos x$. We are looking for the number of solutions to the equation $x^{2}-1 = \cos x$. Observe the behavior of both functions: $1$. The function $f(x) = x^{2}-1$ is an upward-opening parabola with its vertex at $(0, -1)$. $2$. The function $g(x) = \cos x$ is a periodic wave oscillating between $-1$ and $1$. At $x = 0$, $f(0) = -1$ and $g(0) = 1$. Since $f(0) For $|x| > \sqrt{1 + \pi/2} \approx 1.57$, the parabola $x^{2}-1$ grows rapidly and exceeds the maximum value of $\cos x$, which is $1$. By analyzing the graph, the parabola $x^{2}-1$ intersects the curve $\cos x$ at exactly two points, one in the interval $(0, \pi/2)$ and one in the interval $(-\pi/2, 0)$.

The graphs of the polynomial $x^{2}-1$ and $\cos x$ intersect

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