Which word represents the idea of rapid growth or increase?

The correct choice embodies the concept of rapid growth or increase because "proliferate" specifically refers to the swift increase or spread of something, often used in contexts like population growth or the expansion of ideas. It implies not just growth, but a considerable and often exponential rise, portraying a sense of vitality and momentum that aligns with the idea of rapid growth.

In contrast, "reproach" relates to expressing disappointment or criticism, "remonstrate" means to protest or object, often in a reasoned manner, and "demure" describes a modest or reserved demeanor—none of which connote the notion of rapid growth or increase. Therefore, "proliferate" is the most fitting word for the concept of swift enhancement or multiplication.