Answer:
The company have to sell 40 items each week to maximize its profit.
Step-by-step explanation:
The given profit function is
\(p(x)=-0.5x^2+40x-300\) .... (1)
where, p is weekly profit in dollars and x is number of sold items.
In the given function leading coefficient is negative, it means it is a downward parabola and vertex of a down word parabola is point of maxima.
If a parabola is defined as
\(f(x)=ax^2+bx+c\) .... (2)
then the function is maximum at \(x=-\frac{b}{2a}\).
From (1) and (2) we get
\(a=-0.5, b=40, c=300\)
\(x=-\frac{b}{2a}=-\frac{40}{2(-0.5)}=40\)
The given function is maximum at x=40.
Therefore the company have to sell 40 items each week to maximize its profit.
