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Aug 16, 2019 · Positive-definite then your function is strictly convex. Positive semi-definite then your function is convex. A matrix is positive definite when all the eigenvalues are positive and semi-definite if all the eigenvalues are positive or zero-valued.
Let \(I\) be an interval and let \(f, g: I \rightarrow \mathbb{R}\) be convex functions. Prove that \(cf\), \(f + g\), and \(\max \{f, g\}\) are convex functions on \(I\), where \(c \geq 0\) is a constant. Find two convex functions \(f\) and \(g\) on an interval \(I\) such that \(f \cdot g\) is not convex. Answer. Add texts here.
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
5 days ago · A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.
Say we have a graph of the function f(x) = x(x^2 + 1). Find the parts of the graph where the function is convex or concave, and find the point(s) of inflexion. [3 marks] f(x) = x(x^2 + 1) = x^3 + x gives. f''(x) = 6x. f''(x) = 0, when x = 0. f''(x) \textcolor{red}{< 0} when x<0. Here we have a concave section. f''(x) \textcolor{purple}{> 0 ...
A function f is concave if the 2nd derivative f’’ is negative (f’’ < 0). Graphically, a concave function opens downward, and water poured onto the curve would roll off. A function f is convex if f’’ is positive (f’’ > 0). A convex function opens upward, and water poured onto the curve would fill it.
The most popular property of convex functions is the Jensen's Inequality. Here's another way to check if a function is convex. Let \(f : I \to \mathbb{R}\) be a twice differentiable function. Then, \(f\) is convex on \(I\) if and only if \(f''(x) \geq 0 \quad \forall x \in I\)
