That said, the real gain of eating vegetables is probably not that they happen to contain anti-oxidants. Rather, they contain any number of things that are good to us which are hard to quantify individually but has been observed to do us good as a whole. To attribute the positive health effects of something as complex as an apple to a single class of components - such as anti-oxidants - is most likely cargo-cult science in its purest form.
But if your aim is to increase people's health, you're going to do a better job using accurate information.
The initial studies showing anti-oxidants have health benefits have struggled to be replicated, and new studies have shown anti-oxidants may be having negative health impact.
Edit: since a couple of people have asked, here's one fairly good source, there are others if you search. http://www.ncbi.nlm.nih.gov/pubmed/22419320
Here is an article from Bad Science pointing to a Cochrane Collaboration review of the effectiveness of anti-oxidants:
http://www.badscience.net/2008/04/cliff-richard-gloria-hunni...
Michael Shermer, the author of Why People Believe Weird Things, also had an interesting direct experience with nutritional pseudo-science:
http://en.wikipedia.org/wiki/Why_People_Believe_Weird_Things
Probably some variation/inversion of Hanlon's Razor.
Not everyone who wants to see more evidence for every claim and trick is doing that out of sheer denial. There might be some room for the Nirvana fallacy here ("SuperMegaFoodStuff didn't make me into an Adonis in a week, so it's crap"), but in general I'm pro-skepticism.
Antioxidants Could Spur Tumors by Acting on Cancer Gene http://www.sciencemag.org/content/343/6170/477.summary
Supplement Alert: Antioxidants May Protect Cancer Cells Instead Of Protecting Us From Them http://www.forbes.com/sites/daviddisalvo/2014/02/08/suppleme...
That links to a 2008 Cochrane review.
Here's the updated review: http://summaries.cochrane.org/CD007176/antioxidant-supplemen...
My smoothie is made from whole foods, only.
Supplements are a very complicated thing. I can't even venture to say whether they're good or not, based on all the conflicting information.
If consuming anti-oxidants has any effect, it's not necessarily a positive one, since the body uses oxidation for various processes, for example the immune system.
You might be consuming anti-oxidants aiming to benefit your health, yet be negatively impacting your immune system's ability to fight infection and prevent cancer.