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How do you know if a function is concave?
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Dec 21, 2020 · 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 \(x=0\).
Taking the second derivative actually tells us if the slope continually increases or decreases. When the second derivative is positive, the function is concave upward. When the second derivative is negative, the function is concave downward.
There are a number of ways to determine the concavity of a function. If given a graph of f (x) or f' (x), determining concavity is relatively simple. Otherwise, the most reliable way to determine concavity is to use the second derivative of the function; the steps for doing so as well as an example are located at the bottom of the page.
We know that a differentiable function [latex]f^{\prime}[/latex] is decreasing if its derivative [latex]f^{\prime \prime}(x)<0[/latex]. Therefore, a twice-differentiable function [latex]f[/latex] is concave down when [latex]f^{\prime \prime}(x)<0[/latex]. Applying this logic is known as the concavity test.
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
Sep 5, 2015 · Let f: Rn → R. We say that f is concave if for all x, y ∈ Rn and for all λ ∈ [0, 1] we have f(λx + (1 − λ)y) ≥ λf(x) + (1 − λ)f(y). And a function is convex if − f is concave, or f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y). Definition (Quasi-concave/Quasi-convex). Let f: Rn → R.
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.
%20is%20concave?&ck=9742D9D01494DF140CB3CE1BAE848076&idpp=rc&idpview=singleimage&form=rc2idp "How do you know if a function f(x) is concave?1")
%20is%20concave?&ck=F493D715BCB0411F0FD64D5FAD96734E&idpp=rc&idpview=singleimage&form=rc2idp "How do you know if a function f(x) is concave?2")
%20is%20concave?&ck=20C8E204A5AF4040BFA279D4F27CAE83&idpp=rc&idpview=singleimage&form=rc2idp "How do you know if a function f(x) is concave?3")
%20is%20concave?&ck=EB579F1DCFCF18B8091245C309795DAD&idpp=rc&idpview=singleimage&form=rc2idp "How do you know if a function f(x) is concave?4")
%20is%20concave?&ck=BD90E84E5EA3AFF88FD8EB242D3C4FBA&idpp=rc&idpview=singleimage&form=rc2idp "How do you know if a function f(x) is concave?5")
