The revenue function for a sporting goods company is given by R(x)=x⋅p(x) dollars where x is the number of units sold and p(x)=100−0.5x is the unit price. Find the maximum revenue.

Question

The revenue function for a sporting goods company is given by R(x)=x⋅p(x)R(x)=x⋅p(x) dollars where x is the number of units sold and p(x)=100−0.5xp(x)=100−0.5x is the unit price. Find the maximum revenue.

Answer 1

So, R(x) = x [100-0.5x] = 100x - 0.5x² or (-1/2)x²+100x

The graph of that function is a downward opening parabola, so its maximum value occurs at its vertex.

The easiest way to locate the vertex (in this case) is to find the equation of the axis of symmetry which is given by x = -b/2a, where a = -1/2 and b = 100.

Since the vertex lies on the axis of symmetry, the x coordinate of the vertex is also -b/2a. That value is where the maximum revenue occurs. Plug that value into the R function, i.e. find R (-b/2a) and you will have the maximum revenue.

Perform the above calculations then check your work here:

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