Which of the following is true about f(x) = 3 | vedclass

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(C) Given the function $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$. Using the definitions $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \

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$f$ is a strictly increasing function on $R$

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(C) Given the function $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$. Using the definitions $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \frac{e^x + e^{-x}}{2}$: $f(x) = 3 \left( \frac{e^x - e^{-x}}{2} \right) - 2 \left( \frac{e^x + e^{-x}}{2} \right)$ $f(x) = \frac{3e^x - 3e^{-x} - 2e^x - 2e^{-x}}{2} = \frac{e^x - 5e^{-x}}{2}$ Now,find the derivative $f'(x)$: $f'(x) = \frac{d}{dx} \left( \frac{1}{2} e^x - \frac{5}{2} e^{-x} \right) = \frac{1}{2} e^x + \frac{5}{2} e^{-x}$ Since $e^x > 0$ and $e^{-x} > 0$ for all $x \in R$,it follows that $f'(x) > 0$ for all $x \in R$. Therefore,the function $f$ is a strictly increasing function on $R$. Hence,option $C$ is correct.

Which of the following is true about $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$?

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