concave up
When f ‘(x) is increasing, f(x) is
When a curve is described as concave up, it means that the curve is curving in the direction in which it opens upward or upward-facing. In other words, the curve is curving upward at an increasing rate as you move from left to right along the curve.
Visually, a concave up curve looks like a smile shape, with its lowest point at the bottom. Mathematically, this means that the curve has a positive second derivative. The second derivative describes the rate at which the curve is curving; a positive second derivative indicates that the curve is curving upward or opening upward, while a negative second derivative indicates that the curve is curving downward or opening downward.
Examples of functions that are concave up include f(x) = x^2 and g(x) = e^x. In both cases, as x increases, the function increases at an increasing rate, resulting in a concave up curve.
Understanding concavity is important in calculus and optimization problems as it helps identify the behavior of a function and its maximum or minimum points.
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