Question

Given the Production Function Q = 72X + 15X2 - X3, where Q =Output and X=Input

a) What is the Marginal Product (MP) when X = 8?

b) What is the Average Product (AP) when X = 6?

c) At what value of X will Q be at its maximum?

d) At what value of X will Diminishing Returns set in?

Answer 1

Given $72X + 15X^2 - X^3$

a) $MP=\frac {dQ} {dX} =72+30X-3X^2$

When X=8;

$72+30(8)-3(8)^2$

=72+240−192=120

MP=120

b) $AP=\frac {Q} {X} =72+15X-X^2$

When X=6;

AP=72+15(6)−(6×6)=126

AP=126

c) Q is maximum when $\frac {dQ} {dX} =0$

$72+30X-3X^2=0$

$3X^2-30X-72=0$

$X^2-10X-24=0$

$X^2-12X+2X-24=0$

$X(X−12)+2(X−12)=0$

X=12or (X=−2 which is inadmissible)

d) Diminishing returns means, MP is decreasing i.e

$\frac {dMP} {dX} \lt 0$

$\frac {dMP} {dX} =0$

30-9X=0

9X=30

X=3.33

Thus diminishing returns sets is when X>3.33