When f ‘(x) is negative, f(x) is
decreasing
When the derivative f ‘(x) is negative, it means that the function f(x) is decreasing.
Intuitively, this means that as you move along the x-axis from left to right, the values of f(x) are getting smaller. This can be seen graphically as a downward slope.
For example, let’s say we have the function f(x) = -x² + 3x + 2. If we take its derivative, we get f ‘(x) = -2x + 3. When f ‘(x) is negative (i.e. when -2x + 3 < 0), we can solve for x to get x > 3/2. This tells us that the function f(x) is decreasing for x > 3/2.
So, when f ‘(x) is negative, we can conclude that f(x) is decreasing at that point.
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