(4 points) Find all real numbers b and c so that the quadratic polynomial f ( x
ID: 2827887 • Letter: #
Question
(4 points) Find all real numbers b and c so that the quadratic polynomial f (x) = x2 + bx + c takes on both positive and negative values. Give one example of such a pair of real numbers (b, c). Under the condition that the function takes on both positive and negative values, show that the absolute minimum of the function occurs at the midpoint between the two roots.
Let C(x) be the cost of producing x units of a certain good with C(0) = 0. Assume that the graph of C(x) on [0,?) is concave up.
(a) (4 points) Give a detailed argument showing that the average cost of producing x units, defined by A(x) = C(x)/x , is minimized at production level x0, where x0 is the production level for which the average cost A(x) equals the estimate of the marginal cost of production given by the derivative of C(x) (see page 152 in Rogawski).
(b) (2 points) Show that the line through the points (0,0) and (x0,C(x0)) is tangent to the graph of C(x).
Explanation / Answer
1.
for f to take both positive and negative values
b^2-4c >0
eaxmple:
take b = 2. c = 0
f(x) = x^2-2x, roots are x = 0,2
f(1) =-1, f(3) =3
let f(x) ==x^2 +bx +c
min occurs at
f'(x) = 2x+b = 0
=>
x = -b/2
and roots are (-b+sqrt(b^2-4c))/2, (-b-sqrt(b^2-4c))/2
we can see that -b/2 is midway between roots
thus proved
2.
A(x) = C(x)/x
=>
A' = (xC'- C)/x^2 = 0
=>
xC' = C
=>
C' = C/x = A
we know that C'(x0) = A(x0)
=>
A has minimum at x = x0
thus proved
(b)
euation of the line is
xoY = C(xo)X
i.e
Y = [C(x0)/x0]X
we know that
C'(x0 ) = A(x0) = C(x0)/x0
=>
equation of the line is
Y = C'(x0)X
the above line has slope C'(x0) and passes throush (x0, C(xo))
=>
the above line is tangent to the graph of C(x)
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