Answer
$(a)$ See the image below.
$(b)$ More than $100$ (At least $101$)
$(c)$ $x ∈ (279.13, 400)$
Between the range $(280, 400)$
Work Step by Step
$(a)$ See the image above.
$(b)$ As we can see from the graph above, profit becomes positive after $100^{th}$ cooktop. So they gain profit from $101^{st}$.
They have to produce more than $100$ (At least $101$).
$(c)$ We have to solve the inequality that is:
$10x+0.5x^2-0.001x^3-5000>15000$
So, we need region, where the graph is above $y=15000$ line.
$x ∈ (279.13, 400)$
As the amount of cooktops should be whole number, we have the range between $(280, 400)$