Let R be the set of all real numbers and f(x) | vedclass

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(A) Given $f(x) = \sin^{10} x (1 + \cos^2 x + \cos^4 x + \cos^8 x)$.Taking the natural logarithm on both sides:$\ln f(x) = 10 \ln(\sin x) + \ln(1 + \c

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$S = R$

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(A) Given $f(x) = \sin^{10} x (1 + \cos^2 x + \cos^4 x + \cos^8 x)$.Taking the natural logarithm on both sides:$\ln f(x) = 10 \ln(\sin x) + \ln(1 + \cos^2 x + \cos^4 x + \cos^8 x)$.Differentiating with respect to $x$:$\frac{f'(x)}{f(x)} = 10 \cot x + \frac{-2 \cos x \sin x - 4 \cos^3 x \sin x - 8 \cos^7 x \sin x}{1 + \cos^2 x + \cos^4 x + \cos^8 x}$.$\frac{f'(x)}{f(x)} = 10 \cot x - \sin 2x \left[ \frac{1 + 2 \cos^2 x + 4 \cos^6 x}{1 + \cos^2 x + \cos^4 x + \cos^8 x} \right]$.Let $g(x) = \frac{f'(x)}{f(x)}$. For $x \in (0, \pi)$,the term $10 \cot x$ takes all values in $(-\infty, \infty)$.The term involving $\sin 2x$ is a bounded function for $x \in (0, \pi)$.Since the sum of a function with range $(-\infty, \infty)$ and a bounded function is a function with range $(-\infty, \infty)$,the expression $\frac{f'(x)}{f(x)}$ takes all real values $\lambda \in R$ as $x$ varies in $(0, \pi)$.Therefore,$S = R$.

Let $R$ be the set of all real numbers and $f(x) = \sin^{10} x (\cos^8 x + \cos^4 x + \cos^2 x + 1)$ for $x \in R$. Let $S = \{\lambda \in R : \text{there exists a point } c \in (0, 2\pi) \text{ with } f'(c) = \lambda f(c)\}$. Then,

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