Let f(x) = e^x cos x + 1. Which of the follow | vedclass

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(A) Let $f(x) = e^x \cos x + 1$. Let $\alpha$ and $\beta$ be two consecutive roots of $f(x) = 0$ such that $\alpha < \beta$. Then $f(\alpha) = 0$ and

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Between any two consecutive roots of $f(x) = 0$ there is always a root of $e^x \sin x + 1 = 0$

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(A) Let $f(x) = e^x \cos x + 1$. Let $\alpha$ and $\beta$ be two consecutive roots of $f(x) = 0$ such that $\alpha Then $f(\alpha) = 0$ and $f(\beta) = 0$. Since $f(x)$ is continuous on $[\alpha, \beta]$ and differentiable on $(\alpha, \beta)$, by Rolle's Theorem, there exists at least one $c \in (\alpha, \beta)$ such that $f'(c) = 0$. Calculating the derivative: $f'(x) = e^x \cos x - e^x \sin x = e^x(\cos x - \sin x)$. However, consider the function $g(x) = e^{-x} f(x) = \cos x + e^{-x}$. Then $g(\alpha) = 0$ and $g(\beta) = 0$. By Rolle's Theorem, there exists $c \in (\alpha, \beta)$ such that $g'(c) = 0$. $g'(x) = -\sin x - e^{-x} = 0$. Multiplying by $-e^x$, we get $e^x \sin x + 1 = 0$. Thus, there is a root of $e^x \sin x + 1 = 0$ between $\alpha$ and $\beta$.

Let $f(x) = e^x \cos x + 1$. Which of the following statements is always true?

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