Understanding the Decreasing Nature of Functions with Negative Derivatives

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When the derivative of a function f(x), denoted as f'(x), is negative, it means that the function is decreasing in the interval where f'(x) is negative

When the derivative of a function f(x), denoted as f'(x), is negative, it means that the function is decreasing in the interval where f'(x) is negative. In other words, as x increases, the values of f(x) will decrease.

This can be understood by the geometric interpretation of the derivative. The derivative of a function represents the slope of the function at a particular point. When the derivative is negative, it indicates that the slope of the function is negative at that point. This implies that the function is sloping downwards or decreasing.

For example, consider the function f(x) = 2x^2 – 3x + 1. If we take the derivative of this function, we get f'(x) = 4x – 3. Set f'(x) < 0 to find the values of x where the function is decreasing: 4x - 3 < 0 4x < 3 x < 3/4 Thus, when x < 3/4, the derivative f'(x) is negative, indicating that the function f(x) = 2x^2 - 3x + 1 is decreasing in that interval. In summary, when f'(x) is negative, f(x) is decreasing, or the values of f(x) are getting smaller as x increases.

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