Which of the following is true about $f(x)=3$ $\sinh (x)-2 \cosh (x), \forall x \in \mathbf{R}$

Which of the following is true about $f(x)=3$ $\sinh (x)-2 \cosh (x), \forall x \in \mathbf{R}$
  1. $f$ is an odd function
  2. $f$ is a periodic function
  3. $f$ is a strictly increasing function on $\mathbf{R}$
  4. $f$ is a strictly decreasing function on $\mathbf{R}$

Solution

Given, function, $f(x)=3 \sin h(x)-2 \cos h(x)$, $\forall x \in \mathbf{R}$ $ \begin{aligned} & =3\left(\frac{e^x-e^{-x}}{2}\right)-2\left(\frac{e^x+e^{-x}}{2}\right) \\ & =\frac{1}{2} e^x-\frac{5}{2} e^{-x}=\frac{1}{2}\left(e^x-5 e^{-x}\right) \\ \because \quad f^{\prime}(x) & =\frac{1}{2}\left(e^x+5 e^{-x}\right)>0 \forall x \in \mathbf{R} \end{aligned} $ $\therefore$ The, function $f$ is a strictly increasing function on $R$ Hence, option (3) is correct

Asked in: AP EAMCET 2020 (22 Sep Shift 1)

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