PROJECT 5 - OPTIMIZATION
paul vaz
1. Consider the function f ( x)  x 3  4 x 2  4 x,
for 0  x  4.
(a) Locate the critical points.
(It may be a good idea to view the graph of the function before calculating the
critical points)
(b) Find the global maximum and minimum on 0  x  4 .
(See graph)
2. For some positive constant C, the temperature change, T, in a patient generated by a
dose, D, of a drug is given by
C D
T  (  )D 2 .
2 3
(a) What dosage maximizes the temperature change?
(b) The sensitivity of the body, at dosage D, to the drug is defined as dT/dD. What
dosage maximizes sensitivity?
3. A woman pulls a sled which, together with its load, has a mass of m kg. If her arm
makes an angle of  with her body (assumed vertical) and the coefficient of friction
( a positive constant) is , the least force, F, she must exert to move the sled is given
by
mg
F
.
sin    cos
If  = 0.15, find the maximum and minimum values of F for 0     / 2 . Give your
answers as multiples of mg.
Useful Commands:
 k:=expr;
 f:=diff(k,x);
 solve(f=0);
 subs(x=a,k);
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 T:=(((C/2)-(D/3))*D^2);
 h:=diff(T,D);
 j:=factor(h);
 diff(j,D);
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 F:=0.15/(sin(x)+0.15*cos(x));
 F1:=diff(F,x);
 solve(F1=0);
 plot(F,x=0..Pi/2);
 subs(x=a,F);
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