8. The function C(r) given by: C(x) = 2x² +10 + 32 Define the function: AC(r) = C(x)/.r for x > 0. Define the function: MC(r) dC(r)/dc. Define the term "convex function" and prove that C(c) is convex. Plot the AC and MC functions. Use calculus to establish that the AC function has a unique minimum and that the MC function intersects the AC function at that point. C(x) = 2x² + 10x +32 2
8. The function C(r) given by: C(x) = 2x² +10 + 32 Define the function: AC(r) = C(x)/.r for x > 0. Define the function: MC(r) dC(r)/dc. Define the term "convex function" and prove that C(c) is convex. Plot the AC and MC functions. Use calculus to establish that the AC function has a unique minimum and that the MC function intersects the AC function at that point. C(x) = 2x² + 10x +32 2
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:1
5
8. The function C(r) given by:
1
dc
2
C(x) = 2x² +10 + 32
Define the function: AC(r) = C(x)/x for x > 0. Define the function:
MC(c) dC(r)/dc. Define the term "convex function" and prove
that C(c) is convex. Plot the AC and MC functions. Use calculus
to establish that the AC function has a unique minimum and that the
MC function intersects the AC function at that point.
dx
C64) = 2x²³² + 10x +32
AC(x) =
22 +10 +32
x
=
= mclxy) = 4x +10
A function f(x) is convex if f"(x) >
Note
d2c
Jx²=4>0
бас
Ix
= 2-32
бчас
(
dz²
x2
=O
.
mc(4)
so C is convex for
6470
23
The second deviostre test shows that x=4 is a
minimin
So there is
9
= x=4
Note AC 700 as x70,
only
AC (A) = 8 + 10 + 8 = 26
16+10
= 26/
x>0
I minime
(270)
AC7 ∞ as x-200
for
(0) 1 xx
according
to the grat
270.
Samevale so they
meet.
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