Understanding the Negative Slope of f'(x) and its Impact on the Decreasing Nature of f(x)

When f ‘(x) is negative, f(x) is

When f'(x) is negative, it means that the slope or gradient of the function f(x) is negative at that particular point x

When f'(x) is negative, it means that the slope or gradient of the function f(x) is negative at that particular point x. This indicates that as x increases, the function f(x) is decreasing.

In other words, when f'(x) is negative, the function f(x) is decreasing or getting smaller as x increases. This can be seen visually on a graph as a downward slope.

For example, let’s say we have a function f(x) = 3x^2. We take the derivative of f(x) to find f'(x) = 6x. Now, when f'(x) = 6x is negative, it means that x is negative because any negative value multiplied by 6 gives a negative result. Therefore, f(x) is decreasing or getting smaller as x becomes more negative.

In general, when f'(x) is negative, it signifies a decreasing behavior of the function f(x) in the neighborhood of that specific point x.

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