If f(x) is concave down, then f”(x) is?
If a function f(x) is concave down, it means that its graph is shaped like a downward-facing curve
If a function f(x) is concave down, it means that its graph is shaped like a downward-facing curve. In mathematical terms, concave down means that the second derivative of f(x), denoted as f”(x), is negative.
To understand this relationship, let’s consider the definition of concavity and the second derivative test:
1. Concavity: A function f(x) is concave down on an interval if the slope of the function is decreasing as x increases within that interval. In other words, the graph of f(x) is curving downward.
2. Second derivative: The second derivative of a function f(x) represents the rate at which the slope is changing. If the second derivative is positive, it means the slope is increasing, and if it is negative, the slope is decreasing.
Now, if f(x) is concave down, it implies that the slope is decreasing as x increases. This means that the rate at which the slope is changing (i.e., the second derivative, f”(x)) must be negative. In other words, f”(x) < 0. To summarize, if a function f(x) is concave down, then f''(x) is negative.
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