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A function is concave (convex) if the graph of the function is always above (below) any chord (line segment between two points in the graph). Remark 4. f concave ⇔−f convex. Example 5. Let S = [0,∞) and consider f(x) = √ x and g(x) = −f(x) = − √ x f is a concave function and g is a convex function. 1.2 Selected properties of ...
A function $f(x)$ is said to be quasi-concave if its domain and all its $\alpha$-superlevel sets defined as $$\mathcal{S}_\alpha \triangleq\{x|x\in dom f, f(x)\geq\alpha\}$$ are convex for every $\alpha$.
Sep 9, 2023 · Definition: An object or a function is concave if it curves inward. In simple terms, it’s hollow or bowed in, much like a cave. Everyday Examples: A bowl. A satellite dish. A spoon’s interior. Skateboard ramps. A pie with a slice taken out of it. Convex.
In a ray diagram, a concave lens is drawn as a vertical line with inward facing arrows to indicate the shape of the lens. Learn about and revise lenses, images, magnification and...
Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
- You have two functions f and g, where f''>0 and g''>0. If you take the second derivative of f+g, you get f''+g'', which is positive. So their sum i...
- In order for 𝑓(𝑥) to be concave up, in some interval, 𝑓 ''(𝑥) has to be greater than or equal to 0 (i.e. non-negative) for all 𝑥 in that inter...
- In general, no. Points where f"(x)= 0 are specifically called inflection points. But in the example provided, -2 is also a critical point (Observe...
- Product rule: d/dx[f(x)*g(x)] = f'(x)g(x) + f(x)g'(x) f(x) = (x-1)³ and g(x) = x
Nov 14, 2023 · A concave function (sometimes called ''concave down'') is a function for which any secant line (a line connecting two points on the function) lies below the...
Example. Let's see an example of concave function: Commonly, we can say that the convex functions are curved functions that are first decreasing and afterwards increasing, while the concave functions are the other way round, they are first increasing and afterwards increasing.
