Question 258477
The given function to determine the cost per bicycle is ..
C(x)=0.5x^2-1.2x+2.725 .

The standard form of a quadratic equation is ..
f(x)=ax^2+bx+c

Comparing the above two equations we can say that the Cost function C(x) is in a standard quadratic equation form where..

a = 0.5, b = -1.2, c = 2.725

Now if any quadratic equation is plotted on a graph we get a parabola (which looks like the letter "U" or an inverted "U")

For quadratic equations..
if a > 0 , The parabola is in the form of a letter "U" i.e., it opens upwards. This also means that the function has a minimum which is at the vertex of the parabola.

Considering the standard form of quadratic equation, the vertex of the parabola has co-ordinates {-b/2a ,  f(-b/2a)}

So,
 x = -b/2a
 Substituting the values for a and b we can find the value of x.

 x = -(-1.2)/2(0.5) = 1.2/1 = 1.2

So, the Cost function has a minimum when x = 1.2.

Since the Cost function is for 100x bicycles , aki's bicycles needs to manufacture atleast 100(1.2) bicycles to minimize the cost per bicycle.

So the total no. of required bicycles is 120

Verify:
Now lets verify our answer.
The value of x that we got is 1.2. Lets substitute this in the Cost function C(x)

C(x) = 0.5(1.2)^2 - 1.2(1.2) + 2.725 
     = 0.5(1.44) - 1.44 + 2.725 
     = 0.72 + 1.285
     = 2.005

Lets raise the value of x to 2 and substitute in the cost function C(x)

C(x) = 0.5(2)^2 - 1.2(2) + 2.725
     = 0.5(4) - 2.4 + 2.725
     = 2 + 0.325
     = 2.325

So as you can see as we raised the value of x from 1.2 to 2 the C(x) function jumped up from 2.005 to 2.325. In other words what this means is the value of x=1.2 is the minimum for this function.

Hope this helps.