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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.
Aug 16, 2019 · You can check arbitrary points or in case the second derivative exist you want your second derivative be non negative. That makes the slopes to increase and the curve falls above the tangent lines and below the secant lines.
1.4 Examples of multivariate convex functions. A ne functions: f(x) = aT x + b (for any a 2 Rn; b 2 R). They are convex, but not strictly convex; they are also concave: 8 2 [0; 1]; f( x + (1. )y) = aT ( x + (1 )y) + b = aT x + (1. )aT y + b + (1 )b. = f(x) + (1. )f(y): In fact, a.
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his method can be used to prove Jensen’s inequality for convex functions [2, Theorem 90]. Recall a function f: [a;b] !R is said to be convex if f( x+ (1 )y) f(x) + (1 )f(y) for any x;y2[a;b] and any 2[0;1]. Moreover, fis called strictly convex if it is convex and f( x+ (1 )y) = f(x) + (1 )f(y) implies x= yor 2f0;1g.
Convex Functions You are probably familiar with the notion of concavity of functions. Given a twice-di erentiable function ’: R !R, We say that ’is convex (or concave up) if ’00(x) 0 for all x2R. We say that ’is concave (or concave down) if ’00(x) 0 for all x2R. For example, a quadratic function ’(x) = ax2 + bx+ c
A real-valued function f is said to be convex on I if it lies below (or on) the straight-line chords between pairs of points of its graph. In other words, if x1, x2 are points of I with x1 < x2, and ax + b is the linear function agreeing with f(x) at x1 and x2, then f(x) ax + b for x1 x x2.
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The function f is weakly convex if, for every a; b 2 I and every t 2 (0; 1), Terminology: Spivak's \convex" means strictly convex. Many other author's \convex" means weakly convex. Example: the function f(x) = 1=x is convex on (0; 1) and concave on (1 ; 0). Geometric meaning: f is weakly convex i.
