Suppose total benefits and total costs are given by b(y) = 100y − 8y2 and c(y) = 10y2. what is the maximum level of net benefits (rounded to the nearest whole number)?
Whenever you face the problem that deals with maxima or minima you should keep in mind that minima/maxima of a function is always a point where it's derivative is equal to zero. To solve your problem we first need to find an equation of net benefits. Net benefits are expressed as a difference between total benefits and total cost. We can denote this function with B(y).
B(y)=b-c B(y)=100y-18y²
Now that we have a net benefits function we need find it's derivate with respect to y.
[tex] \frac{dB(y)}{dy} =100-36y[/tex]
Now we must find at which point this function is equal to zero.
0=100-36y 36y=100 y=2.8
Now that we know at which point our function reaches maxima we just plug that number back into our equation for net benefits and we get our answer.
B(2.8)=100(2.8)-18(2.8)²=138.88≈139.
One thing that always helps is to have your function graphed. It will give you a good insight into how your function behaves and allow you to identify minima/maxima points.
