Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = xe⁻(ˣ²/₂)

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To determine where the function f(x) = x * e^(-x²/2) 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). Here, u(x) = x and v(x) = e^(-x²/2).

Differentiate u(x) = x to get u'(x) = 1. For v(x) = e^(-x²/2), use the chain rule. The derivative of e^u is e^u * u', where u = -x²/2. So, v'(x) = e^(-x²/2) * (-x).

Apply the product rule: f'(x) = u'(x) * v(x) + u(x) * v'(x) = 1 * e^(-x²/2) + x * (-x) * e^(-x²/2). Simplify this to get f'(x) = e^(-x²/2) - x² * e^(-x²/2).

Factor out e^(-x²/2) from f'(x): f'(x) = e^(-x²/2) * (1 - x²). The sign of f'(x) depends on the sign of (1 - x²) since e^(-x²/2) is always positive.

Solve the inequality 1 - x² > 0 to find where f'(x) > 0 (increasing) and 1 - x² < 0 for f'(x) < 0 (decreasing). The solution to 1 - x² > 0 is -1 < x < 1, so f(x) is increasing on (-1, 1). For 1 - x² < 0, x < -1 or x > 1, so f(x) is decreasing on (-∞, -1) and (1, ∞).

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Critical Points

Critical points are values of x in a function where the derivative is either zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. To find critical points, we first compute the derivative of the function and set it equal to zero, solving for x.

First Derivative Test

The First Derivative Test is a method used to determine the behavior of a function at its critical points. By analyzing the sign of the derivative before and after each critical point, we can conclude whether the function is increasing or decreasing in those intervals. If the derivative changes from positive to negative, the function is decreasing; if it changes from negative to positive, the function is increasing.

Exponential Functions

Exponential functions, such as f(x) = xe⁻(ˣ²/₂), involve a constant raised to a variable exponent. In this case, the function combines polynomial and exponential components, which can affect its growth and decay rates. Understanding the behavior of exponential functions is crucial for analyzing their derivatives and determining intervals of increase and decrease.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.f(x) = xe⁻(ˣ²/₂)