The cost of manufacturing x units of a commodity is 5x² + 30x + 5. Find the range of output where…

Average cost $AC=\frac{5x^2+30x+5}{x}=5x+30+\frac{5}{x}$. $\frac{d(AC)}{dx}=5-\frac{5}{x^2}$. AC is increasing when $5-\frac{5}{x^2}>0$, i.e. $x^2>1$, so $x>1$ (output cannot be negative).