MHT CET · Maths · Application of Derivatives
The set of all points, for which \(\mathrm{f}(x)=x^2 \mathrm{e}^{-\dot{x}}\) strictly increases, is
- A \((0,2)\)
- B \((2, \infty)\)
- C \((-2,0)\)
- D \((-\infty, \infty)\)
Answer & Solution
Correct Answer
(A) \((0,2)\)
Step-by-step Solution
Detailed explanation
\(\mathrm{f}^{\prime}(x)=2 x \mathrm{e}^{-x}-x^2 \mathrm{e}^{-x}=x \mathrm{e}^{-x}(2-x)\)
Since f is increasing, \(\mathrm{f}^{\prime}(x)\gt0\)
\(\begin{aligned}
& \Rightarrow x \mathrm{e}^{-x}(2-x)\gt0 \\
& \Rightarrow x(2-x)\gt0 \\
& \Rightarrow 0 \lt x \lt 2 \\
& \Rightarrow x \in(0,2)
\end{aligned}\)
Since f is increasing, \(\mathrm{f}^{\prime}(x)\gt0\)
\(\begin{aligned}
& \Rightarrow x \mathrm{e}^{-x}(2-x)\gt0 \\
& \Rightarrow x(2-x)\gt0 \\
& \Rightarrow 0 \lt x \lt 2 \\
& \Rightarrow x \in(0,2)
\end{aligned}\)
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