Question

# Is the vector space C∞[a,b] of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite inte

Transformations of functions

Is the vector space $$C \infty[a,b]$$ of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by $$D(f)(x)=f′(x)D(f)(x)=f′(x)\ and\ I(f)(x)=∫xaf(t)dt f(t)dt$$. (a) Compute $$(DI)(f)=D(I(f))(DI)(f)=D(I(f))$$. (b) Compute $$(ID)(f)=I(D(f))(ID)(f)=I(D(f))$$. (c) Do this transformations commute? That is to say, is it true that $$(DI)(f)=(ID)(f)(DI)(f)=(ID)(f)$$ for all vectors f in the space?

a) $$(DI)(f)=f(x)$$
b) $$(ID)(f)=f(x)-f(a)$$
c) $$(DI)(f)=(ID)(f)$$ only when $$f(a)=0$$