Understanding Decreasing Functions: the Relationship Between the Derivative and Decreasing Values

f'(x)<0

When the derivative of a function, denoted as f'(x), is less than zero (f'(x) < 0), it indicates that the function is decreasing

When the derivative of a function, denoted as f'(x), is less than zero (f'(x) < 0), it indicates that the function is decreasing. In other words, as x increases, the values of f(x) decrease. This means that the slope of the function is negative, and the graph of the function has a downward slope. The function is decreasing in the interval where this condition is satisfied. To better understand this concept, consider the graph of a simple linear function. Let's say the function is given by f(x) = mx + b, where m is the slope and b is the y-intercept. If the slope m is negative (m < 0), then the function is a line that starts at a higher value of f(x) and decreases as x increases. For example, let's take the function f(x) = -2x + 5. The derivative of this function is f'(x) = -2. Since the derivative is negative (f'(x) < 0), it means that the function is decreasing. As x increases, the values of f(x) decrease. This can be observed visually on the graph of the function, where the line has a downward slope. In summary, if f'(x) < 0, it means that the function f(x) is decreasing. This indicates that as x increases, the values of f(x) decrease.

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