#29 p231 show that f(x)=a*x^3+b*x^2+c*x+d has 2 or 0 extrema!
var('a,b,c,d')
f(x)=a*x^3+b*x^2+c*x+d
g(x)=diff(f(x),x)
show(f(x))
show(g(x))
show(solve(g(x)==0,x))
#when b^2-3ac>0 f'(x)--0 has 2 real roots, so f(x) has 2 extrema!
#when b^2-3ac<0, f'(x)==0 has 2 complex roots, so f(x) has 0 extrema!!
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#23 p231 maximize profit given revenue, r(x)=8*sqrt(x) and cost, c(x)=2*x^2
r(x)=8*sqrt(x)
c(x)=2*x^2
p(x)=r(x)-c(x)
rp(x)=diff(r(x),x)
cp(x)=diff(c(x),x)
pp(X)=diff(p(x),x)
show(r(x))
show(c(x))
show(p(x))
show(rp(x))
show(cp(x))
show(pp(x))
show(solve(pp(x)==0,x)[2])
show(p(1))
p1=plot(p(x),.1,4,color='red')
p2=plot(pp(x),.1,4,color='green')
#since p(x)=r(x)-c(x), p'(x)=r'(x)-c'(x)
#so p(x) has a min or a max when r'(x)=c'(x)!
#max occurs at x=1 (thousand widgets)
#and max = p(1) = 6 (mega bux)
p1+p2
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#20) minimize the time T(x)=t1(x)+t2(x) it takes to row to a village V
#20) if you are 2 miles off shore in a boat B and the village is 6 miles from the point closest to the shore S
#20) and you row 2mph but walk 5mph! Let L be the point of land fall, LS=x, VS=6, VL=6-x and SB=2.
#20) V========================L=========S
#20) d1=6-x + x +
#20) + +
#20) + +
#20) + +
#20) d2 + + 2
#20) + +
#20) + +
#20) + +
#20) ++
#20) B
d1(x)=sqrt(x^2+4)
d2(x)=6-x
t1(x)=d1(x)/2
t2(x)=d2(x)/5
T(x)=t1(x)+t2(x)
Tp(x)=diff(T(x),x)
show(T(x))
show(Tp(x))
show(solve(Tp(x)==0,x))
show(solve(25*x^2==4*x^2+16,x)[1].rhs().n())
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