f(x) is concave up when f’ is ____ or f” is ____
increasing, f” > 0
To determine the concavity of a function, we need to analyze the second derivative of the function f(x).
If f”(x) > 0 for all values of x, then f(x) is concave up (convex) on its entire domain. This means that the slope of the tangent line to the graph of f(x) is increasing as x increases.
Alternatively, we can also check the concavity of f(x) by analyzing the sign of its first derivative f'(x). If f'(x) is increasing, then f(x) is concave up. If f'(x) is decreasing, then f(x) is concave down.
If f(x) is a differentiable function, then f”(x) > 0 implies that f'(x) is increasing. Hence, we can also conclude that if f”(x) > 0 for all values of x, then f(x) is both concave up and f'(x) is increasing.
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