f(x) is concave down on an interval, if f'(x) is
To determine whether a function f(x) is concave down on an interval, we need to look at the behavior of its first derivative, f'(x)
To determine whether a function f(x) is concave down on an interval, we need to look at the behavior of its first derivative, f'(x).
If f'(x) is decreasing on the interval, then f(x) is concave down on that interval.
To understand why this is the case, let’s recall the definition of concavity. A function is concave down if its graph “opens downward” or if it curves downward. This means that as we move from left to right on the graph, the slope is decreasing.
The first derivative, f'(x), represents the slope of the function f(x) at any given point. If f'(x) is decreasing on an interval, it means that as we move from left to right on that interval, the slope is consistently getting smaller. In other words, the rate of change of the function is decreasing.
Since a decreasing derivative implies a decreasing slope, it means that f(x) is curving downward or is concave down on that interval.
In summary, if the first derivative, f'(x), is decreasing on an interval, then the function f(x) is concave down on that interval.
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