The short-run total cost curve of the firm is given as SRTC=3q^2/K+4q+432K where SRTC, q, and K represent the short-run total cost, the quantity of output, and the capital input respectively and K is fixed and q is 144. Therefore, the short-run marginal cost of the firm or SRMC=dSRTC/dq=6q/K+4+432K and the short-run average cost or SRAC function of the firm =SRTC/q=(3q^2/K+4q+432K)/q=3q/K+4+432K/q and hence, the derivative of the SRAC=dSRAC/dq=3/K-432K/q^2. Now, based on the optimal or efficient scale of production in the long-run, the firm would operate in the long-run at the point where the SRAC is at its minimum level.

Therefore, based on the efficient or optimal production level in the long-run, it can be stated:-

3/K-432K/q^2=0

(3q^2-432K^2)/Kq^2=0

3q^2-432K^2=0

-432K^2=-3q^2

-432K^2=-3(144)^2 [Plugging the value of q as given in the question]

-432K^2=-62,208

K^2=-62,208/-432

K^2=144

K*=12

Hence, the efficient or optimal level of K in the long-run is 12 when the firm produces 144 units of output implying that the given statement is true, in this instance.

Now, considering that the K is 11 in the short-run and the firm produces 144 units of output, the level of K that would generate the minimum average cost of production of the firm in the long-run would be 12, in this case. This essentially signifies that when the firm hires or employes 12 units of K in the long-run, it would operate at the minumum efficient scale of production or would be able to achieve the lowest possible average cost in the long-run to produce the desired level of output or q , in this case. Therefore, the firm does not have any economic incentive to invest more on K or hire more K beyond or above 12 units as doing so could further increase its average cost of production in the long-run. Hence, the given statement here is false.

Therefore, as the K is fixed at 11 units in the short-run and the long-run efficient or optimal level of K for the firm is 12 units, it can increase the level of K in the production in the long-run by 1 more unit to achieve the optimal and efficient level of K that would minimize the average cost of production in the long-run. If the firm keeps hiring 11 units of K in the long-run it would still be experiencing economies of scale and hence, could employ more amount of K up to 12 units in order to minimize its average cost of production in the long-run and achieve the minimum efficient scale of production. Therefore, the given statement here would be false.

As the efficient or optimal level of K in the long-run is 12 units and the short-run level or amount of K is fixed at 11 units, the firm can increase K by 1 more unit in the long-run to achieve the minimum efficient scale of production. Hence, the statement, in this case, would be false.