If x in (0, 1),then what type of function is | vedclass

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(A) Given the function $f(x) = x^{100} + \sin x - 1$. To determine the nature of the function,we find its derivative $f'(x)$. $f'(x) = \frac{d}{dx}(x^

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Monotonically increasing

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(A) Given the function $f(x) = x^{100} + \sin x - 1$. To determine the nature of the function,we find its derivative $f'(x)$. $f'(x) = \frac{d}{dx}(x^{100} + \sin x - 1) = 100x^{99} + \cos x$. For $x \in (0, 1)$,we observe that $x^{99} > 0$,so $100x^{99} > 0$. Also,for $x \in (0, 1)$,$\cos x > 0$ because $x$ is in the first quadrant. Since both terms $100x^{99}$ and $\cos x$ are positive for all $x \in (0, 1)$,their sum $f'(x) = 100x^{99} + \cos x > 0$. Since $f'(x) > 0$ for all $x \in (0, 1)$,the function $f(x)$ is monotonically increasing on the interval $(0, 1)$.

If $x \in (0, 1)$,then what type of function is $f(x) = x^{100} + \sin x - 1$?

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