If f(x) is concave up, then f”(x) is?
f”(x) = positive
If a function f(x) is concave up, then it means that its graph is shaped like a smile with an upward curve. This implies that the slope of its tangent line increases as we move towards the right.
Mathematically, if f(x) is concave up, then we know that its second derivative f”(x) is positive (i.e., f”(x) > 0) for all x in the domain of f(x). This is because the second derivative of a function is the rate of change of its slope, and a positive second derivative means that the slope is increasing as we move towards the right.
Conversely, if f”(x) is positive for all x in the domain of f(x), then it implies that the graph of f(x) is concave up. This is a result of the fact that the second derivative test can tell us about the concavity of a function.
In summary, if f(x) is concave up, then f”(x) is positive, and if f”(x) is positive, then f(x) is concave up.
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