Increasing returns to scale

Question

Let $f=f(K,L)$ - production function. Let $f(1,2) \le4 $ and a production function is defined to have increasing returns to scale. What maximum product can the firm produce using K=3 and L=6?

So we know that $f(3,6) = f(3*1, 3*2) > 3*f(1,2)$ and $f(1,2)\le4$

But what should I do next?

Answer 1

What I believe you mean is that a production function has Constant Returns to Scale.

i.e. $$f(\lambda K, \lambda L)= \lambda f(K, L) $$

Where $\lambda \ge 0$

Thus in your case where $K=3$ and $L=6$

$$f(3,6)=3 \times f(1,2)$$

and since ${f_{max}(1,2)=4}$ (i.e. $f(1,2) \le4$) the maximum product a firm can produce using $K=3$ and $L=6$ would be

$$3 \times f(1,2)= 3 \times 4 =12$$

Hope this helps!