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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
Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
Nov 21, 2023 · Examples of concavity: Consider the function {eq}f(x)=\frac{1}{8} x^4-3x^2 {/eq}. The first derivative would be {eq}f' (x)=\frac{1}{2} x^3-6x {/eq}.
If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points .
If f ′ (x) is negative on an interval, the graph of y = f(x) is decreasing on that interval. The second derivative tells us if a function is concave up or concave down. If f ″ (x) is positive on an interval, the graph of y = f(x) is concave up on that interval.
Dec 21, 2020 · The canonical example of \(f''(x)=0\) without concavity changing is \(f(x)=x^4\). At \(x=0\), \(f''(x)=0\) but \(f\) is always concave up, as shown in Figure \(\PageIndex{11}\). Figure \(\PageIndex{11}\): A graph of \(f(x) = x^4\). Clearly \(f\) is always concave up, despite the fact that \(f''(x) = 0\) when \(x=0\).
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A function $f(x)$ is said to be concave if the two conditions below satisfied: (i) $dom f$ is convex. (ii) For all $x,y\in dom f$ and $\theta\in[0,1],$ $$f(\theta x+(1-\theta)y) \geq \theta f(x) + (1-\theta)f(y)$$ In fact, this implies that if a function is convcave then that's also quasi-concave but not necessarily the converse is true.
