AP EAMCET · Maths · Application of Derivatives
If \(f(x)=x e^{x(1-x)}, x \in \mathbb{R}\), then \(f(x)\) is
- A increasing on \(\left[-\frac{1}{2}, 1\right]\)
- B decreasing on \(\mathbb{R}\)
- C increasing on \(\mathbb{R}\)
- D decreasing on \(\left[-\frac{1}{2}, 1\right]\)
Answer & Solution
Correct Answer
(A) increasing on \(\left[-\frac{1}{2}, 1\right]\)
Step-by-step Solution
Detailed explanation
\(f'(x) = \frac{d}{dx}\left(x e^{x-x^2}\right) = 1 \cdot e^{x-x^2} + x \cdot e^{x-x^2} \cdot (1-2x)\) \(f'(x) = e^{x-x^2} (1 + x - 2x^2)\) \(f'(x) = -e^{x-x^2} (2x^2 - x - 1)\) \(f'(x) = -e^{x-x^2} (2x+1)(x-1)\) For \(f(x)\) to be increasing, \(f'(x) > 0\).…
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