If f(x) is concave down, then f”(x) is?
If a function f(x) is concave down, this means that the graph of the function is curved in a way that resembles a “smile”
If a function f(x) is concave down, this means that the graph of the function is curved in a way that resembles a “smile”. In other words, the second derivative of the function, denoted as f”(x), must be negative.
To understand why this is the case, let’s recall the definition of concavity. A function is concave down if a line segment connecting any two points on the graph of the function lies entirely below the graph. This implies that the slope of the line connecting any two points on the graph must be decreasing as you move from left to right along the graph.
The slope of a function can be determined by its first derivative. If f'(x) is decreasing, then its first derivative, denoted as f”(x), must be negative. On the other hand, if f'(x) is increasing, then f”(x) would be positive.
Therefore, if the function f(x) is concave down, then f”(x) must be negative. Conversely, if f”(x) is negative, it indicates that the function is concave down.
More Answers:
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Understanding Concavity in Calculus: Identifying the Relationship Between the Second Derivative and Graph Shape