Understanding Concavity in Math | Explaining the Relationship Between a Function’s Curvature and its 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 curving upwards

If a function f(x) is concave up, it means that the graph of the function is curving upwards. In other words, if you were to draw a smooth curve that represents the function, it would be shaped like a “U” (opening upwards) or some portion of it.

The second derivative of a function, denoted as f”(x), represents the rate at which the slope of the function is changing. Specifically, it tells us whether the function is curving upwards (concave up) or curving downwards (concave down).

So, if f(x) is concave up, then f”(x) is positive. This is because a positive second derivative indicates that the slope of the function is increasing, resulting in a concave up shape. Conversely, if f”(x) is negative, it means that the function is concave down.

To summarize:

– If f(x) is concave up, then f”(x) > 0 (positive).
– If f(x) is concave down, then f”(x) < 0 (negative).

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