# AKI'S Bicycle Designs has determined that when x hundred bicycles are built, the average cost per...

## Question:

AKI'S Bicycle Designs has determined that when {eq}x {/eq} hundred bicycles are built, the average cost per bike is given by

{eq}C(x) = 0.1x^2 - 1.4x + 10.766 {/eq},

where {eq}C(x) {/eq} is in hundreds of dollars.

How many bicycles should the shop build to minimize the average cost per bicycle?

## Minimum of a Function:

The minimum point of a function is the point where its derivative takes the value zero. However, as the maximum point also has a slope of zero, we use the second derivative test to make sure the point we have found is a minimum.

The average cost will be minimum where the derivative of the average cost function is equal to zero. This point is found as follows.

\begin{align} C'(x)& =\frac{\mathrm{d} }{\mathrm{d} x} \left (0.1x^2 - 1.4x + 10.766 \right )\\ &=2*0.1x^{2-1}-1.4x^{1-1}+0\\ &=0.2x-1.4\\ &\text{Finding the minimum point,} \\&0.2x-1.4=0\\ \therefore x&=7 \end{align}

The second derivative of the function is:

\begin{align} C"(x)& =\frac{\mathrm{d} }{\mathrm{d} x} \left (0.2x-1.4\right )\\ &=0.2 \end{align}

As the second derivative is always negative, the point we have found is the minimum point.

As x is in hundreds, the average cost will be minimum when 700 bicycles are produced.