Answer to Question #209208 in Calculus for Akhona Klanisi

Question

Question 15

A firm’s production function is given by

Q = 700Le^−0,02L,

where Q denotes the number of units produced and L the number of labourers. Determine the size of the  workforce that maximises output.

[a.   L = 14

[b.   L = 50

[c.   L = 328

[d.   L = 700

Answer 1

$$Q(L) = 700Le^{−0.02L}$$

$$Q'(L)=(700Le^{−0.02L})'=700e^{−0.02L}(1-0.02L)$$

Critical number(s):

$$Q'(L)=0=>700e^{−0.02L}(1-0.02L)=0$$

$$1-0.02L=0$$

$$L=50$$

If $L<50, Q'(L)>0, Q(L)$ increases.

If $L>50, Q'(L)<0, Q(L)$ decreases.

The function $Q(L)$ has a local maximum at $L=50.$

Since the function $Q(L)$ has the only extremum, then the function $Q(L)$ has the absolute maximum at $L=50.$

[b.   L = 50