The cost of manufacturing x units of a commodity is $27 + 15x+3x^2$. Find the output for which AC is decreasing or increasing.

AC is decreasing for $0<x<3$ and increasing for $x>3$.

The correct answer is Option (1) → AC is decreasing for $0<x<3$ and increasing for $x>3$.

$C(x)=27 + 15x+3x^2$

$∴AC=\frac{C}{x}=\frac{27}{x}+15+3x$

$\frac{d}{dx}(AC)=-\frac{27}{x^2}+3$

AC decreases when $\frac{d}{dx}(AC)<0$

$⇒3-\frac{27}{x^2}<0⇒3<\frac{27}{x^2}⇒1<\frac{9}{x^2}$

$⇒ x^2 < 9$ (Multiplying both sides of inequality by $x^2$, which is +ve)

$⇒ x^2-9 <0⇒ (x+3) (x-3) < 0⇒ -3 < x < 3$.

As x is +ve, AC decreases in the range $0 < x < 3$.

AC increases when $\frac{d}{dx}(AC)>0$

$⇒3-\frac{27}{x^2}>0⇒1>\frac{9}{x^2}⇒ x^2 > 9$

$⇒ x^2-9 >0⇒ (x+3) (x-3) > 0⇒ x < -3$ or $x>3$

As x is +ve, AC increases when $x > 3$.