15) Show that the fune [6] Let f : (a, b) → R be strictly convex on (a,b). Show that there is 80 cE (a, b) such that...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Convex Optimization Let f: R R be a differentiable function on R. Show that f is convex iff f' is nondecreasing (i.e. x y f'(x) <f'(y)).
(4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation for the minimizer u.(x) of the following problem: minimize u subject to: u E A, where A- 0,REC1[0 , 1and u (0 a u(1)b) b) Assuming the minimizer u(a) is a C2 function, prove t is strictly convex (4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation for the minimizer u.(x)...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
: R → Rn be a Ci path which solves the 1. Let F : Rn → R be a C1 function, and let differential equation, E'(t)--VF(C(t)), te R. (a) Show that f(t) F((t)) is a non-increasing function of t (ie, f'(t) 30 Vt.) (b) For any t for which F(E(t)) * 0, decreasing in t (ie, f'(t) <0.) show that č is a smooth path, and f(t) is strictly
7. [8 POINTS] Let f: R → R be a strictly increasing function. Prove by way of contradiction that there cannot be more than one place where f crosses the x-axis.
need help with all a, b, c 2. 15 Marks (a) Suppose that f : R" R is convex but not necessarily smooth. Prove that h-af is a (b) Suppose that f : R -R is convex and smooth. Also assume that f(x) > 0 for all z (c) Show that the set S = {(x,y) : y > 0} is convex and that the function f(x,y)-x2/v is convex function if a-0. Show with a simple example that this is...
1.1 Be f: R->R given by , Show that f ist convex 1.2 Be f: given by . Show that f ist convex 1.3 Show, that for all applies : f(x) (0.00) → R Oo f(r) =-In(2) We were unable to transcribe this imageInla f(x) (0.00) → R Oo f(r) =-In(2) Inla
Exercise 5.3.4: Let f: [a,b] → R be a continuous function. Let ce [a,b] be arbitrary. Define po the Prove that F is differentiable and that F'(x) = f(x) for all x € [a,b]. series on the
6. Let f [a, b R be a thrice differentiable function and xo E [a, b]. Show that da 6. Let f [a, b R be a thrice differentiable function and xo E [a, b]. Show that da