# Least cost

• August 3rd 2006, 03:10 PM
bret80
Least cost
A producer of accesories has production costs described by the function:

C= 800 - 10x + 1/4*x^2

x is the number of accesories produced per day.

Find the number of accesories needed to be produced on a daily basis in order to have the least cost.

a) 10
b) 20

Through substitution 20 seems to be the right answer, but is there other way to know?
• August 3rd 2006, 03:15 PM
topsquark
Quote:

Originally Posted by bret80
A producer of accesories has production costs described by the function:

C= 800 - 10x + 1/4x^2

x is the number of accesories produced per day.

Find the number of accesories needed to be produced on a daily basis in order to have the least cost.

a) 10
b) 20

Through substitution 20 seems to be the right answer, but is there other way to know?

There are three ways. You can graph the function, you can put the quadratic in the general form of a parabola (which gives you the vertex, the lowest point on the parabola) or, since you are in Calculus you can take the first derivative of the cost function and set it equal to zero:

$C= 800 - 10x + 1/4x^2$

$C' = -10 + (1/2)x = 0$

Thus
$-10 + (1/2)x = 0$
$(1/2)x = 10$
$x = 20$.

To verify this is actually a relative minimum and not a relative maximum, apply the second derivative test:
$C'' = 1/2$.
Since C'' is positive at x = 20 (in fact C'' is positive everywhere) x = 20 is a relative minimum.

-Dan