The function $f(x)=xe^{-3x}$ :

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To determine the behavior of the function $ f(x) = xe^{-3x} $, we need to analyze its derivative, which will help us understand where the function is increasing or decreasing. 1. **Find the derivative of $ f(x) $:** The function $ f(x) = xe^{-3x} $ is a product of two functions: $ x $ and $ e^{-3x} $. We will use the product rule to find the derivative: $$ f'(x) = \frac{d}{dx}(x) \cdot e^{-3x} + x \cdot \frac{d}{dx}(e^{-3x}) $$ $$ f'(x) = 1 \cdot e^{-3x} + x \cdot (-3e^{-3x}) $$ $$ f'(x) = e^{-3x} - 3xe^{-3x} $$ $$ f'(x) = e^{-3x}(1 - 3x) $$ 2. **Determine where $ f'(x) > 0 $ (increasing) and $ f'(x) < 0 $ (decreasing):** The sign of $ f'(x) = e^{-3x}(1 - 3x) $ depends on the factor $ (1 - 3x) $, since $ e^{-3x} $ is always positive for all real $ x $. - $ f'(x) > 0 $ when $ 1 - 3x > 0 $, which simplifies to $ x < \frac{1}{3} $. - $ f'(x) < 0 $ when $ 1 - 3x < 0 $, which simplifies to $ x > \frac{1}{3} $. Therefore, the function $ f(x) $ is increasing on the interval $ (-\infty, \frac{1}{3}) $ and decreasing on the interval $ (\frac{1}{3}, \infty) $. 3. **Explanation of why Option 3 is correct:** Option 3 states that the function increases in $ (-\infty, \frac{1}{3}) $. This is consistent with our analysis of the derivative, which shows that $ f'(x) > 0 $ for $ x < \frac{1}{3} $. Thus, the function is indeed increasing on this interval, making Option 3 the correct choice. 4. **Explanation of why the other options are incorrect:** - **Option 1: Increases on R** - This option is incorrect because the function does not increase over the entire real line $ R $. It only increases on the interval $ (-\infty, \frac{1}{3}) $ and decreases for $ x > \frac{1}{3} $. - **Option 2: Decreases on R** - This option is incorrect because the function does not decrease over the entire real line $ R $. It increases on $ (-\infty, \frac{1}{3}) $ and only decreases for $ x > \frac{1}{3} $. - **Option 4: Decreases in $(-∞,\frac{1}{3})$** - This option is incorrect because, as shown, the function actually increases in the interval $ (-\infty, \frac{1}{3}) $, not decreases. In conclusion, the analysis of the derivative confirms that Option 3 is the correct answer, as it accurately describes the interval where the function $ f(x) = xe^{-3x} $ is increasing.