A firm's average cost function is: $$AC = x^2 – 2x + 50 , \ 0 \leq x$$
a) Find the level of output to minimalize average cost (AC) function
b) Find the level of output when average cost equals marginal cost (AC=MC)
Could you please check my calculations ? I have done a) and a little b):
a)$$AC = x^2 – 2x + 50$$
$$f'(x)=2x-2=0$$
$$x-1=0$$ $$x=1$$
$$f(1) = 1 – 2 + 50 = 49$$
b) This one I'm totally not sure
$$TC=AC·x$$
$$TC=x^3 – 2x^2 + 50x$$
$$MC=TC'=3x^2-4x+50$$
$$AC=MC$$
$$x^2 – 2x + 50=3x+2-4x+50$$
$$x_1=1$$$$x_2=0$$
Best Answer
The average cost function is the parabola for $x\ge 0$ $$\operatorname{AC}(x)=(x-1)^2+49$$ which has the minimum at the point $P=(1,49)$.
The total cost function is $\operatorname{TC}(x)=\operatorname{AC}(x)\cdot x$ and then differentiating $$ \underbrace{\operatorname{TC}'(x)}_{\operatorname{MC}(x)}=\operatorname{AC}'(x)\cdot x+\operatorname{AC}(x) $$ and using $\operatorname{MC}(x)=\operatorname{AC}(x)$ and $\operatorname{AC}'(x)=2(x-1)$ we have $$ 0=\operatorname{AC}'(x)\cdot x=2(x-1)x $$ that is $x=0$ or $x=1$.