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
$f^{\prime}(x)>0$
$\Rightarrow 1-\mathrm{e}$
$x>0$
$x<0$
$x \in(-\infty, 0)$