Let \( C(x) \) and \( R(x) \) be the cost function and the revenue function for \( x \) units of a commodity such that marginal cost of producing 2 units is equal to marginal revenue of selling 1 unit. If \( C(x)=ax^3+bx^2-12x+6 \) and \( R(x)=100+6ax-bx^2 \), then which one of the following is TRUE?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( a+b-2=0 \)
Marginal cost and marginal revenue are the derivatives of C(x) and R(x):
\( \displaystyle MC=C'(x)=3ax^2+2bx-12 \)
\( \displaystyle MR=R'(x)=6a-2bx \)
\( \displaystyle MC\ \text{at}\ x=2:\ 12a+4b-12 \)
\( \displaystyle MR\ \text{at}\ x=1:\ 6a-2b \)
Equating them:
\( \displaystyle 12a+4b-12=6a-2b \)
\( \displaystyle 6a+6b=12 \)
\( \displaystyle a+b-2=0 \)