How is antioxidant activity of carotenoids determined?
As Buck Thorn and I commented elsewhere, the antioxidant activity of carotenoids has been determined using $\ce{DPPH}$ radical ($\ce{DPPH^.}$)(Refs 1 &amp; 2). According to Ref.1, antioxidant activities of the isolated α- and β-carotene have been analyzed in five different concentrations of 0.001, 0.005, 0.01, 0.05, and 0.1% ($w/v$) using several assays: The metal chelation, $\ce{DPPH}$ radical scavenging, and superoxide scavenging activities. All of these assays have been showed almost similar results in terms of high activity at lowest concentrations. Thus, it is not true that $\ce{DPPH}$ radical assay cannot be use to measure β-carotene free radical inhibition. 

For latest information on $\ce{DPPH}$ radical assay, read Ref.3. In general, following is a typical procedure for the assay:


  The $\ce{DPPH}$ assaying of $\beta$-carotene can be performed using known antioxident (e.g., Trolox, ascorbic acid, etc.) as reference compound. The assay can be carried out in either alcoholic media (e.g., methanol or ethanol) or in an aprotic solvent (e.g., acetonitrile). To the degassed solutions ($\pu{2 mL}$) of antioxidant (with a final concentration from $0$ to $\pu{60 μM}$), were added $\pu{2 mL}$ of $\pu{140 μM}$ $\ce{DPPH^.}$ stock solution (70 μM $\ce{DPPH^.}$ final concentration). The solution was stirred at room temperature in the absence of light, and, after $\pu{60 min}$, the absorbance was measured at $\pu{517 nm}$ using an UV-vis spectrometer. The working $\ce{DPPH^.}$ concentration was calculated from the absorbance response at $\pu{517 nm}$ of the initial solution in the absence of antioxidant. The $\ce{DPPH^.}$ scavenging percentage or inhibition ($I\%$) was calculated from the measured absorbance using the following expression:
  
  $$I\% = \frac{A-A_0}{A_0}\times 100$$
  Where $A$ and $A_0$ represent the absorbance at a given antioxidant concentration and the absorbance in the absence of given antioxidant, respectively.  


References:


Lars Müller, Kati Fröhlich, Volker Böhm, “Comparative antioxidant activities of carotenoids measured by ferric reducing antioxidant power (FRAP), ABTS bleaching assay (αTEAC), DPPH assay and peroxyl radical scavenging assay,” Food Chem. 2011, 129(1), 139−148 (https://doi.org/10.1016/j.foodchem.2011.04.045).
Surashree Sen Gupta1, Mahua Ghosh, “In Vitro Antioxidative Evaluation of $\alpha$- and $\beta$-Carotene, Isolated from Crude Palm Oil,” Journal of Analytical Methods in Chemistry 2013, Article ID 351671 (10 pages) (http://dx.doi.org/10.1155/2013/351671).
Om P. Sharma, Tej K. Bhat, “DPPH antioxidant assay revisited,” Food Chem. 2009, 113(4), 1202−1205 (https://doi.org/10.1016/j.foodchem.2008.08.008).
The most recent reference for $\ce{DPPH}$ assay: Magda M. Becker, Gilvanda S. Nunes, Danilo B. Ribeiro, Francisco E. P. S. Silva, Gaëlle Catanante, Jean-Louis Marty, “Determination of the Antioxidant Capacity of Red Fruits by Miniaturized Spectrophotometry Assays,” Journal of the Brazilian Chemical Society 2019, 30(5), 1108-1114 (http://dx.doi.org/10.21577/0103-5053.20190003).