Question

a manufacturer knows that if x goods are demanded on a particular week, the total cost and revenue function will be: c[x]=14+3x and R[x]=18x-2x^2 respectively. calculate the level of demand that will maximize profits. calculate the amount of profit that will be realized at this maximum point

Answer 1

i)

profit function:

$$P(x)=𝑅(𝑥)-𝐶(𝑥)=18x-2x^2-(14+3x)$$

$$=-2x^2+15x-14, x\geq 0$$

Differentiare with respect to $x$

$$P'(x)=(-2x^2+15x-14)=-4x+15$$

Find the critical number(s)

$$P'(x)=0=>-4x+15=0=>x=15/4$$

If $0\leq x\leq15/4, P'(x)>0, P(x)$ increases.

If $x>15/4, P'(x)<0, P(x)$ decreases.

The profit function $P(x)$ has a local maximum at $x=15/4.$

Sinse the profit function $P(x)$ has the only extremum, then the function $P(x)$ has the absolute maximum at $x=15/4.$

Level of demand that will maximize profits:

$$x=15/4=3.75$$

If $x$ is an integer, then

$$P(3)=-2(3)^2+15(3)-14=13$$

$$P(4)=-2(4)^2+15(4)-14=14$$

We take $x=4.$

ii)

The amount of profit that will be realized at this maximum point:

$$P_{max}(x)=P(3.75)$$

$$=-2\cdot(3.75)^2+15\cdot3.75-14=14.125$$

If $x$ is an integer, then

$$P_{max}(x)=P(4)$$

$$=-2\cdot(4)^2+15\cdot4-14=14$$