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In calculus, a derivative represents how a function changes as its input changes. It’s like measuring how fast a car speeds up or slows down. For the function \( f(x) = x^2 e^{-2x} \), finding the derivative gives you insight into its behavior. Here, we need to use some calculus tools like the product rule, because we have two functions multiplied: \( x^2 \) and \( e^{-2x} \).

The derivative, represented as \( f'(x) \), is calculated to determine the function's slope at any point. Positive values of \( f'(x) \) mean the function is increasing, and negative values mean it's decreasing. By calculating \( f'(x) \), you can see when the function is going up or down, which is essential for plotting graphs or solving real-world problems.

Understanding when a function increases or decreases involves examining the sign of its first derivative. This is key in analyzing the behavior of functions. If \( f'(x) > 0 \), the function is increasing at that portion of \( x \). Conversely, if \( f'(x) < 0 \), the function is decreasing.

In the exercise, we analyzed the sign of \( f'(x) = 2xe^{-2x}(1-x) \) across different intervals based on critical points \( x = 0 \) and \( x = 1 \). By testing values like \( x = -1 \), \( x = 0.5 \), and \( x = 2 \), we discovered the behavior in each interval:

• \( (-\infty, 0) \): The function decreases.
• \( (0, 1) \): The function increases.
• \( (1, \infty) \): The function decreases.

This test helps create a "map" of how the function grows and shrinks, which is pivotal in telling the complete story of a function's graph.

Critical points are values of \( x \) where the first derivative \( f'(x) \) is zero or undefined. These points are crucial because they often indicate where a function's graph stops increasing and starts decreasing, or vice versa.

For \( f(x) = x^2 e^{-2x} \), setting the first derivative to zero ordered us to solve \( 2xe^{-2x}(1-x) = 0 \). We found two main critical points: \( x = 0 \) and \( x = 1 \). These critical points typically correspond to peaks, valleys, or transitions in a graph, making them essential for understanding how the graph behaves throughout its range. By identifying these points, you can better visualize and analyze the function's overall shape.

The product rule is a derivative rule that is essential when differentiating functions that are multiplied together. If you have two functions, \( u \) and \( v \), the product rule states that the derivative of their product is \( (uv)' = u'v + uv' \).

For the function \( f(x) = x^2 e^{-2x} \), we used the product rule to find \( f'(x) \):

• Set \( u = x^2 \) and \( v = e^{-2x} \).
• Find \( u' = 2x \) and use the chain rule to find \( v' = -2e^{-2x} \).
• Apply the formula: \( f'(x) = (x^2)' e^{-2x} + x^2 (e^{-2x})' = 2x e^{-2x} - 2x^2 e^{-2x} \).

This rule is fundamental for tackling problems where you encounter the product of two separate functions, giving you the power to accurately analyze rate of change.