Question

The interval of increase of the function f(x) = x − e^x + tan (2π/7) is
(a) (0, ∞)
(b) (−∞, 0)
(c) (1, ∞)
(d) (−∞, 1)

Answer 1

(b) (−∞, 0)

$$\begin{gathered} f\left(x\right)=x-e^x+\tan \left(\frac{2\mathrm{\pi }}{7}\right) \\ f\text{'}\left(x\right)=1-e^x \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow 1-e^x>0 \\ \Rightarrow e^x<1 \\ \Rightarrow x<0 \\ \Rightarrow x\in \left(-\infty ,0\right) \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is increasing on}\left(-\infty ,0\right). \end{gathered}$$