we have to proved that the two given set are convex when function f is convex
f(x) Sa} 5. Show that if f is a convex function on R" then for any...
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)).
Let S function, f: S R, between the two sets. x < 1}. Show that S and R have the same cardinality by constructing a bijective x E R 0
x < π Find the Fourier series representation of the function f (x)-1 over the interval-r
Find any global max or global min ) For the function f(x) = 2x3 - 6x2 +6 ;(-1<x<3)
(5) (5pt) Show that if a convex function f as two distinct global minima r, r** ina convex set K, then it has infinitely many minima. (5) (5pt) Show that if a convex function f as two distinct global minima r, r** ina convex set K, then it has infinitely many minima.
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a, b] by some a < b in R. Does there always exist a differentiable function F:RR such that F' = f? Provide either a counterexample or a proof. (b) Assume that f is differentiable, f'(x) > 1 for every x ER, f(0) = 0. Show that f(x) > x for every x > 0. (c) Assume that f(x) = 2:13 + x. Show that...
Given f(x) = x2 - 6x +5, x<3 Graph f(x) and f-?(x) to show that they are inverses of one another.
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.