Mark the correct alternative in the following:
Question:

Mark the correct alternative in the following:

The interval of increase of the function $f(x)=x-e^{x}+\tan \left(\frac{2 \pi}{7}\right)$ is

A. $(0, \infty)$

B. $(-\infty, 0)$

C. $(1, \infty)$

D. $(-\infty, 1)$

Solution:

Formula:- The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$

Given:-

$f(x)=x-e^{x}+\tan \left(\frac{2 \pi}{7}\right)$

$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=1-\mathrm{e}^{\mathrm{x}}=\mathrm{f}^{\prime}(\mathrm{x})$

Now

$f^{\prime}(x)>0$

$\Rightarrow 1-\mathrm{e}$

$x>0$

$x<0$

$x \in(-\infty, 0)$

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