1. **Problem Statement:** We want to find the smallest perimeter of a rectangle given that its area is 12 cm². 2. **Define variables:** Let $x$ be the length of one side of the rectangle. Then the other side length is $\frac{12}{x}$ because area $= x \times \frac{12}{x} = 12$. 3. **Perimeter formula:** The perimeter $P$ of a rectangle is given by $$P = 2(\text{length} + \text{width}) = 2\left(x + \frac{12}{x}\right).$$ 4. **Function to minimize:** Since minimizing $P$ is equivalent to minimizing the expression inside the parentheses, the function to minimize is $$f(x) = x + \frac{12}{x}.$$ 5. **Check the options:** - $f(x) = \frac{x^2}{12}$ is not related to perimeter. - $f(x) = 2x + \frac{24}{x}$ is twice the perimeter, so minimizing it is equivalent but not the simplest form. - $f(x) = 12x^2$ is unrelated. - $f(x) = x + \frac{12}{x}$ is the correct function to minimize. 6. **Conclusion:** The function to minimize to find the smallest perimeter is $$f(x) = x + \frac{12}{x}.$$