Q229 / 270 · NDA 2015 (Phase 2) · Mathematics
The function $f(x)=\frac{x^2}{e^x}$ monotonically increasing if
- A$\mathrm{x} \lt 0$ only
- B$x \gt 2$ only
- Correct answerC$0 \lt \mathrm{x} \lt 2$
- D$\mathrm{x} \in(-\infty, 0) \cup(2, \infty)$
Solution
$\begin{aligned}
& f(x)=\frac{x^2}{e^x} \\
& f^{\prime}(x)=\frac{2 x \cdot e^x-e^x \cdot x^2}{\left(e^x\right)^2} \\
& f^{\prime}(x)=\frac{2 x-x^2}{e^x}
\end{aligned}$
as $\mathrm{e}^{\mathrm{x}}$ is always positive and for monotonically increasing; $2 x-x^2 \gt 0$
$\begin{array}{ll}
\Rightarrow & \mathrm{x}^2-2 \mathrm{x} \lt 0 \\
\Rightarrow & \mathrm{x}(\mathrm{x}-2) \lt 0 \\
\Rightarrow & \mathrm{x}=(0,2)
\end{array}$
Hence, the correct answer is option: 'C'.