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Introduction. We will now see how derivatives affect the concavity of the graph, but before that, what does the word concavity actually mean? Well, the concavity is actually in a
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The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below. Concave up.
Dec 21, 2020 · Definition Concave Up and Concave Down. Let \(f\) be differentiable on an interval \(I\). The graph of \(f\) is concave up on \(I\) if \(f'\) is increasing. The graph of \(f\) is concave down on \(I\) if \(f'\) is decreasing. If \(f'\) is constant then the graph of \(f\) is said to have no concavity.
Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
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The mathematical definition of a function being concave between points $x_1$ and $x_2$ is the following: $\lambda f(x_1)+(1-\lambda)f(x_2) \leq f(\lambda x_1+(1-\lambda)x_2)$, for any $0 \leq \lambda \leq 1$. Can someone give a detailed, intuitive explanation of this theorem?
