# Marginal revenue, and maximizing revenue & average revenue

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##### Intros
###### Lessons
1. Marginal Revenue, and Maximizing Revenue & Average Revenue Overview:
2. Understanding and Maximizing Marginal Revenue
3. Understanding and Maximizing Average Revenue
##### Examples
###### Lessons
1. Finding & Maximizing Revenue
Given the following information, find the marginal revenue and the value of $q$ which maximizes the revenue:
1. $R(q)=-q^3+4q+2$
2. $R(q)=-\frac{200}{q^2} -2q$
3. $p=- \frac{1}{20} q+100$
4. $q= \frac{50-p}{2}$
2. Finding & Maximizing Average Revenue
Given the following information, find the marginal average revenue and the value of $q$ which maximizes the average revenue:
1. $R(q)=-3q^4+18q^2+5q$
2. $R(q)=-2q^2-20$
3. $p= - \frac{1}{10} q+25$
4. $q= \frac{100-p}{5}$
###### Topic Notes
Marginal Revenue (MR) is the additional revenue that is gained when you increase the unit by one. It is also the derivative of the revenue function. In other words,
$MR=R'(q)$

Average Revenue (AR) is the amount of revenue generated per unit. In other words,
$AR(q)=\frac{R(q)}{q}$
In this section, we would want to find the quantity $q$, which maximizes revenue and average revenue. To maximize revenue, we would want to solve for:

$MR=0$

To maximize average revenue, we would want to solve for:

$AR'(q)=0$