Final answer:
The statement that is NOT true for all three transport mechanisms (facilitated diffusion, primary active transport, and secondary active transport) is that they operate without the need for membrane proteins.
Step-by-step explanation:
Facilitated diffusion does not require energy input since it is a form of passive transport, where substances move down their concentration gradient with the help of carrier proteins. On the other hand, both primary active transport and secondary active transport involve the use of energy.
All three processes involve carrier proteins, but the key distinction is the energy requirement. Facilitated diffusion does not require energy, whereas primary and secondary active transport do require energy input, but they differ in how they source that energy.