Understanding Concavity in Math: Exploring the Relationship between Function Shape and Second Derivative

If f(x) is concave up, then f”(x) is?

If a function f(x) is concave up, it means that the graph of the function is shaped like a “U” or a “smile” and opens upwards

If a function f(x) is concave up, it means that the graph of the function is shaped like a “U” or a “smile” and opens upwards. This implies that the rate at which the function is increasing is getting faster as x increases.

We can determine the concavity of a function by looking at its second derivative, denoted as f”(x). The second derivative provides information about the curvature of the graph.

If f”(x) is positive, it means that the graph of f(x) is concave up. This is because positive values of f”(x) indicate that the function is increasing at an accelerated rate, resulting in a U-shaped graph.

To summarize:
– If f(x) is concave up, then f”(x) is positive.
– If f(x) is concave down, then f”(x) is negative.

Remember, this relationship between concavity and the sign of the second derivative holds for twice-differentiable functions. For functions that are not twice-differentiable, this relationship may not hold.

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