Q.A firm's revenue function is $R(x) = 100x - 2x^2$.

Use the vertex formula for the maximum, and factorise for the zeros; the zeros should be symmetric about the vertex as a check. …

1. Output at maximum revenue. $R(x) = -2x^2+100x$ has $a=-2$, $b=100$. Since $a<0$, the parabola opens downward and has a maximum at

   $x = -\frac{b}{2a} = -\frac{100}{2(-2)} = -\frac{100}{-4} = 25$

   Maximum revenue:

   $R(25) = 100(25) - 2(25)^2 = 2500 - 2(625) = 2500-1250 = ₹1{,}250$

2. Output levels at which revenue is zero. $100x - 2x^2 = 0 \ \Rightarrow \ 2x(50-x) = 0 \ \Rightarrow \ x=0 \ \text{or} \ x=50$ …

Since the roots are 0 and 50 (from part ii), the vertex's x-value must be their midpoint, (0+50)/2 = 25 — the same answer obt …

Forgetting to double the −2 in the denominator of the vertex formula (writing −100/2 instead of −100/(2×−2)). Also, factorising 100x − 2x² incorrectly, e.g. as 2x( …