3.The monthly income from the sale of x units is given by: R(𝑥) = 12𝑥 − 0.01𝑥^2 Dollars Determine the number of units that must be sold each month to maximize income. How much is the maximum income?

To maximize the monthly income, we need to find the vertex of the parabola represented by the equation:

R(x) = 12x - 0.01x^2

The general form of a quadratic equation is:

R(x) = ax^2 + bx + c

For the given equation:

a = -0.01

b = 12

c = 0

The vertex form of a quadratic equation is given by:

x = -\frac{b}{2a}

Substitute a and b into the vertex form equation:

x = -\frac{12}{2(-0.01)}

x = -\frac{12}{-0.02}

x = 600

Thus, the number of units that must be sold each month to maximize income is:

x = 600

Next, substitute x = 600 back into the original equation to find the maximum income:

R(600) = 12(600) - 0.01(600)^2

R(600) = 7200 - 0.01 \times 360000

R(600) = 7200 - 3600

R(600) = 3600

Therefore, the maximum income is:

R(600) = 3600