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Critical points are where we need to focus when finding the minimum values of functions. They are points where the derivative of a function (the slope) is zero, or where the derivative does not exist. These points are crucial because they can indicate where a function might be shifting from an increasing to a decreasing behavior, or vice versa.
To find critical points for the function \( f(x) = x e^x \), we first find its derivative and set it to zero. The derivative is \( f'(x) = e^x(1 + x) \). Setting \( f'(x) = 0 \), we get the equation \( e^x (1 + x) = 0 \). Since \( e^x \) is never zero, we solve the equation \( 1 + x = 0 \), finding \( x = -1 \).
This critical point is essential because it is a candidate for the minimum value of the function within the given interval.

The derivative of a function helps us understand how the function behaves—whether it increases or decreases and where these changes happen. For any given function \( f(x) \), its derivative \( f'(x) \) shows the rate of change of \( f(x) \) with respect to \( x \). To find the derivative of \( f(x) = x e^x \), we use product rule: \( f'(x) = e^x + x e^x \), simplified to \( f'(x) = e^x(1 + x) \).
When the derivative is zero (\