On the range of the derivatives of a smooth function between Banach spaces

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C ^p smooth (Fréchet smooth or a continuous Gâteaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gâteaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous Gâteaux smooth function f from X to Y, with bounded support, so that f′(X) = script L sign(X, Y). In the Fréchet case, we get that if a Banach space X has a Fréchet smooth bump and dens X = dens script L sign(X, Y), then there is a Fréchet smooth function f: X → Y with bounded support so that f′(X) = script L sign(X, Y). Moreover, we see that if X has a C ^p smooth bump with bounded derivatives and dens X = dens script L sign _s ^m(X; Y) then there exists another C ^p smooth function f: X → Y so that f ^(k)(X) = script L sign _s ^k(X; Y) for all k = 0, 1, ..., m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X*. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.