an appliance company determines that in order to sell x dishwashers in a year, they should use a unit price…

Explanation:

Step1: Define the revenue function

Revenue $R$ is the product of the number of units sold $x$ and the unit - price $p$. Given $p = 620-0.4x$, then $R(x)=x\cdot p=x(620 - 0.4x)=620x-0.4x^{2}$.

Step2: Find the derivative of the revenue function

The derivative of $R(x)$ with respect to $x$ is $R^\prime(x)=\frac{d}{dx}(620x - 0.4x^{2})$. Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $R^\prime(x)=620-0.8x$.

Step3: Set the derivative equal to zero to find critical points

Set $R^\prime(x) = 0$. So, $620-0.8x=0$. Solving for $x$: [ \begin{align*} 0.8x&=620\ x&=\frac{620}{0.8}\ x& = 775 \end{align*} ]

Step4: Confirm it's a maximum

The second - derivative of $R(x)$ is $R^{\prime\prime}(x)=\frac{d}{dx}(620 - 0.8x)=-0.8<0$. Since $R^{\prime\prime}(x)<0$ when $x = 775$, the function $R(x)$ has a maximum at $x = 775$.

Answer:

775