when f'(x) is negative
When f'(x) is negative, it means that the derivative of the function f(x) is negative for all values of x in its domain
When f'(x) is negative, it means that the derivative of the function f(x) is negative for all values of x in its domain. In other words, the slope of the tangent line to the graph of f(x) is negative at every point on the graph.
This information tells us that as x increases, the values of f(x) are decreasing. In terms of the graph of f(x), it means that the function is “falling” or “decreasing” as we move from left to right. The steepness of this decrease is given by the magnitude of the negative value of f'(x).
To further understand this concept, let’s consider an example:
Example: Let f(x) = x^2 – 3x. Determine when f'(x) is negative.
To find f'(x), we need to differentiate f(x) with respect to x:
f'(x) = d/dx (x^2 – 3x)
= 2x – 3
Now, we want to find when f'(x) is negative:
2x – 3 < 0
To solve this inequality, we isolate x:
2x < 3
x < 3/2
Therefore, f'(x) is negative for all values of x less than 3/2. This means that the function f(x) = x^2 - 3x is decreasing for all x values less than 3/2.
In summary, when f'(x) is negative, it indicates that the function f(x) is decreasing or "falling" as x increases.
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