Identify the open intervals on which the function is increasing or decreasing.
$f(x)=xe^{-2x}$

• A. Increasing on $\left(-\infty,\frac12\right)$, Decreasing on $\left(\frac12,\infty\right)$
• B. Increasing on $\left(\frac12,\infty\right)$$\left(-\infty,\frac12\right)$, Decreasing on $\left(-\infty,\frac12\right)$
• C. Increasing on $\left(-\infty,\infty\right)$
• D. Decreasing on $\left(-\infty,\infty\right)$

Answer: (A)

Verified step by step guidance

1. To determine where the function f(x) = x * e^(-2x) is increasing or decreasing, we first need to find its derivative, f'(x). Use the product rule for differentiation, which states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x).
2. Identify u(x) = x and v(x) = e^(-2x). Compute the derivatives: u'(x) = 1 and v'(x) = -2 * e^(-2x) (using the chain rule for the derivative of e^(-2x)).
3. Apply the product rule: f'(x) = 1 * e^(-2x) + x * (-2 * e^(-2x)) = e^(-2x) - 2x * e^(-2x).
4. Factor out e^(-2x) from the expression for f'(x): f'(x) = e^(-2x) * (1 - 2x).
5. Determine the critical points by setting f'(x) = 0: e^(-2x) * (1 - 2x) = 0. Since e^(-2x) is never zero, solve 1 - 2x = 0 to find x = 1/2. Analyze the sign of f'(x) on the intervals (-∞, 1/2) and (1/2, ∞) to determine where the function is increasing or decreasing.