University of Missouri Columbia Advanced Calculus Quiz


I’m working on a Calculus drill and insufficiency buttress.

1. (5 pts) Prove that for anyx >0, y >0(x2+y2)12≥(x3+y3)13.Hint: Introduce a new variablez= (yx)2.2. (5 pts) Letf, gbe consistent functions on [0,1], differentiable on(0,1) and such that|f′(x)|<2|g′(x)|for everyx∈(0,1).Which of thefollowing are practicable?(A)f(0) =−5, f(1) = 0, g(0) =−4, g(1) = 0(B)f(0) =−5, f(1) = 0, g(0) =−4, g(1) =−1(C)f(0) = 5, f(1) = 0, g(0) =−4, g(1) =−2(D)f(0) = 5, f(1) = 0, g(0) =−4, g(1) =−33. (5 pts) Prove that the functionf(x) =x+x2sin2xifx6= 0,f(0) = 0Prove thatfhas actual derivative at 0. Isfincreasing on any meantime(−ε, ε),ε >0?