Answer to Question #271054 in Calculus for keneth

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

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

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