vv

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These are answers submitted by vv


 

The Maple answer is correct for your question. It gives the two intervals on which the derivative is negative.
If you are interested in the (relative) extrema, use exprema.
Of course x=0 is not an extreme point.

A := x^2 + 400/x;

x^2+400/x

(1)

plot(A, x=-10..10, view=-500..500);

 

solve(evalf(diff(A, x) < 0));lprint([%]);

RealRange(-infinity, Open(0.)), RealRange(Open(0.), Open(5.848035476))

 

[RealRange(-infinity, Open(0.)), RealRange(Open(0.), Open(5.848035476))]

 

solve(diff(A, x) < 0);lprint(%);  # it is better not to approximate A

RealRange(-infinity, Open(0)), RealRange(Open(0), Open(2*5^(2/3)))

 

RealRange(-infinity, Open(0)), RealRange(Open(0), Open(2*5^(2/3)))

 

extrema(A, {}, x, 'S'); evalf(%);S=evalf(S);

{60*5^(1/3)}

 

{102.5985568}

 

{{x = 2*5^(2/3)}} = {{x = 5.848035476}}

(2)

 

Download extrema.mw

restart;
eq1:= f(x) + 2*f(1/x) + 3*f(x/(x-1)) = x:
eq:=eq1, simplify(eval(eq1, x=1/x)):
eq:=eq, simplify(eval(eq1, x=x/(x-1))):
eq:=eq, simplify(eval(eq1, x=-1/(x-1))):
eq:=eq, simplify(eval(eq1, x=1-x)):
eq:=eq, simplify(eval(eq1, x=(x-1)/x)):
solve([eq], indets([eq],function)): f(x) = eval(f(x), %);
simplify( eval(eq1,  f = unapply(rhs(%), x)) );   # check

             

             

 

So, you want the surface integral on the projection of a cuboid on a plane.

V:=<0,0,0>, <1,2,0>, <0,2,0>, <1,0,0>, <0,0,1>, <1,2,1>, <0,2,1>, <1,0,1>:
T:=<-1,0,0>, <0,-1,0>, <0,0,-1>: 
PM:=LinearAlgebra:-ProjectionMatrix([T[2]-T[1], T[3]-T[1]]):
Pr := v -> T[1]+PM.v:
Pr2:= v -> (PM.v)[[1,2]]:  # ConvexHull works in dimension 2.
P:=Pr~([V]):
P2:=Pr2~([V]):
ind:=ComputationalGeometry:-ConvexHull(Matrix(P2)^+):
C:=P[ind]: n:=nops(C):  # the vertices of the projection
Tri:=(v1,v2,v3) -> Surface( v1 + t*(v2+s*(v3-v2)-v1), s=0..1, t=0..1):
# the parametrization of the interior of a triangle
add(VectorCalculus:-SurfaceInt( x^2+y^2, [x,y,z] = Tri(C[1],C[i],C[i+1])), i=2..n-1);

                         

Maple solved your equation with respect to z.
Note that LambertW is a function just like sin or exp.

with(VectorCalculus):
SurfaceInt( x^4+y^4+z^4, [x,y,z] = Sphere( <0,0,0>, a ) );

(12/5)*Pi*a^6

(1)

# You may want to see the generated double integral
SurfaceInt( x^4+y^4+z^4, [x,y,z] = Sphere( <0,0,0>, a ), inert );

Int(Int(a^6*(2*cos(theta)^4*cos(phi)^4-4*cos(theta)^4*cos(phi)^2-2*cos(theta)^2*cos(phi)^4+2*cos(theta)^4+4*cos(theta)^2*cos(phi)^2+2*cos(phi)^4-2*cos(theta)^2-2*cos(phi)^2+1)*sin(phi), phi = 0 .. Pi), theta = 0 .. 2*Pi)

(2)

 

Just increase Digits (exp(-200) is very small), e.g.

Digits := 20;

 

1. A faster method is to use Iterator.

with(Iterator):
C:=Combination(5,3):
K:=[1,2,3,4,a]:
seq( K[[seq(c[]+1)]], c=C);

        [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [1, 2, a], [1, 3, a], [2, 3, a], [1, 4, a], [2, 4, a], [3, 4, a]

 

2. To print the body of a procedure, use

interface(verboseproc=2);
print(combinat:-subsets);

or, 

showstat(combinat:-subsets);

 

evalf( Int(h, Omega, epsilon=1e-4, method=_CubaCuhre));

gives 4.12122491078804

(To have a probability you forgot probably to divide by (2*Pi)^3.)

You cannot write like this. diff(z, x)  simplifies to 0, because z is interpreted as an independent variable (just like x).
If z depends on x, you must write z(x)
In this case,
diff(w(x,y,t,z(x)), x); 
==>  D[1](w)(x, y, t, z(x)) + D[4](w)(x, y, t, z(x))*diff(z(x), x) 

You may use e.g.
alias(z=z(x));
so that the argument x is omitted, but you must be careful (I personally do not use this facility in such situations).

 

You have a small mistake. Use:

Note that here the origin is at the minimal point of the catenary.

restart;

DIV := diff(y(x), x, x) = a*sqrt(1 + diff(y(x), x)^2):
RV := y(0) = 0, D(y)(0) = 0:
dsolve({DIV, RV}, y(x));

y(x) = (-cosh(a*x)+1)/a, y(x) = (cosh(a*x)-1)/a

(1)

Y:=unapply(rhs(%[2]),x);
# Equation of the catenary, the origin in the minimal point

proc (x) options operator, arrow; (cosh(a*x)-1)/a end proc

(2)

Int(sqrt(1+(D(Y)(u))^2),u=0..x) assuming x>0,a>0

Int((1+sinh(a*u)^2)^(1/2), u = 0 .. x)

(3)

L:=unapply(value(%),x) assuming x>0,a>0;
#Length of the arc (0,0) .. (x,f(x))

proc (x) options operator, arrow; sinh(a*x)/a end proc

(4)

solve([L(x1)=60, Y(x1)=50]);

{a = 1/11, x1 = 11*ln(11)}

(5)

y(x)=eval(Y(x),%);

y(x) = 11*cosh((1/11)*x)-11

(6)

 

You have a linear ordinary differential equation of order 1. The solution contains an integral. This integral, in most cases, cannot be expressed in terms of elementary functions.
Note that in your example, if e.g. theta is a negative integer, the solution is elementary.

# You must provide correct assumptions
restart;
with(inttrans):
#with(gfun);
#assume(alpha + beta = 1):
f := a*t^(-alpha) + b*t^(-beta):
inttrans[laplace](f, t, s) assuming alpha<1,beta<1;

      

This cannot be simplified, even if alpha+beta=1

P:=proc(pmax::posint)
   local p,q,u,v,n,sn, U, R, nR:=0;
   for q to pmax-1 do for p from q+1 to pmax do
     n:=p!/q!; sn:=isqrt(n);
     U:=NumberTheory:-Divisors(n);
     for u in U while u<sn do
       v:=n/u;
       if issqr((v-u)/2) then  nR:=nR+1; R[nR]:=[p, q, (u+v)/2, isqrt((v-u)/2)] fi;
     od:
   od od;
  print('num'=nR);
  entries(R, nolist)
  end proc:

P(13);
                            num = 56
[4, 1, 5, 1], [5, 1, 11, 1], [7, 1, 71, 1], [8, 1, 524, 22],  [8, 1, 204, 6], [8, 1, 201, 3], [9, 1, 684, 18], 
[11, 1, 7911, 69], [10, 1, 2304, 36], [3, 2, 2, 1], [12, 1, 22716, 78], [6, 2, 19, 1], [6, 2, 21, 3], 
[7, 2, 51, 3], [7, 2, 131, 11], [5, 3, 6, 2], [6, 3, 11, 1], [10, 2, 1620, 30], [11, 2, 4905, 45], [10, 3, 916, 22], 
[11, 3, 2956, 38], [7, 3, 29, 1], [8, 3, 104, 8], [8, 4, 76, 8], [13, 3, 32444, 62], [12, 3, 8935, 5], 
[12, 3, 12060, 90], [9, 4, 209, 13], [9, 4, 284, 16], [8, 4, 41, 1], [8, 4, 44, 4], [11, 5, 601, 13], 
[12, 5, 2511, 39], [13, 5, 10449, 87], [10, 6, 71, 1], [9, 4, 124, 4], [12, 4, 4905, 45], [8, 5, 31, 5], 
[9, 5, 55, 1], [11, 6, 236, 4], [11, 6, 244, 8], [11, 7, 336, 18], [8, 7, 3, 1], [11, 6, 431, 19], 
[11, 6, 2316, 48], [11, 6, 249, 9], [11, 6, 276, 12], [13, 8, 659, 23], [12, 9, 61, 7], [9, 8, 5, 2], 
[12, 8, 109, 1], [11, 7, 89, 1], [13, 7, 1231, 23],  [11, 7, 191, 13], [11, 7, 96, 6], [13, 9, 131, 1]
 


 

restart;
Digits:=10:
eq1:=5*(2.477538002*10^(-6)*a^2*b + 1.238769001*10^(-6)*b^3 - 1.761804802*10^(-10)*b)*sin(Upsilon) + 4.081436800*10^(-9)*a - 0.00001717419154*a^3:
eq2:=((2.642707202*10^(-10)*a*sigma + 3.320825870*10^(-8)*a + 1.0*(-3.716307003*10^(-6)*a*sigma - 0.0004669911380*a)*b^2 - 1.858153501*10^(-6)*a^3*sigma - 0.0002334955689*a^3)*sin(Upsilon) + 1.280000000*10^(-12)*(5.026400000*a*sigma + 631.6174240*a)*cos(Upsilon) + 1.280000000*10^(-12)*(1.352400000*sigma^2 + 339.8851680*sigma + 21354.98511)*b + 1.280000000*10^(-12)*(-8.114400000*sigma^4 - 4078.62201638042*sigma^3 - 768779.463909262*sigma^2 - 6.440321829*10^7*sigma - 2.023227102*10^9)*b^3)*csgn(sigma + 125.6600000) - 7.680000000*10^(-13)*(sigma + 125.6600000)^2*b:
eq3:=((1.238769001*10^(-6)*sigma + 0.0001556637127)*b^4 + ((2.477538002*10^(-6)*sigma - 0.0001556637126)*a^2 - 1.761804802*10^(-10)*sigma - 2.213883913*10^(-8))*b^2 - 0.0002334955690*a^4 + 3.320825870*10^(-8)*a^2)*cos(Upsilon) - 8.084703030*10^(-10)*a^2*sin(Upsilon) + (-0.00007783185636 - 2.477538002*10^(-6)*sigma)*a*b^3 + ((0.0003113274252 - 1.238769001*10^(-6)*sigma)*a^3 + (-1.106941957*10^(-8) + 1.761804802*10^(-10)*sigma)*a)*b + (1.286758400*10^(-9)*sigma^2 + 1.616940605*10^(-7)*sigma)*a*b:
sys:=simplify(10^6*[eq1,eq2,eq3]):

nops(sys), indets(sys,name);

3, {Upsilon, a, b, sigma}

(1)

sysr:=convert(sys, rational):

sysr1:=[  eval(sysr,[cos(Upsilon)=c, sin(Upsilon)=d, sigma=1/2])[],c^2+d^2-1 ];

[((66014/5329)*a^2*b+(33007/5329)*b^3-(1303/1479165)*b)*d+(1446/354287)*a-(109932/6401)*a^3, (-(617200709/1316416)*b^2*a+(8637242633/259062403668)*a-(6236490737/26603392)*a^3)*d+(21748019689/26793596995347)*a*c-(469710676743047585773/178516074749979120)*b^3+(73542825103094222810181170291200379/4797756380074395585228714044250000000)*b, ((8927965739/57126880)*b^4+(-(3081170479/19952544)*a^2-8769913225/394562518493)*b^2-(342538/1467)*a^4+(3487/105004)*a^2)*c-(551/681534)*a^2*d-(312269509/3949248)*a*b^3+((8874890483/28563440)*a^3+(26777240151757673/381510694941406250)*a)*b, c^2+d^2-1]

(2)

G:=Groebner:-Basis(sysr1, plex(c,d,a,b)):

nops(G);

8

(3)

indets~(G);

[{b}, {a, b}, {a, b}, {a, b, d}, {a, b, d}, {a, b, c, d}, {a, b, c, d}, {c, d}]

(4)

ab:=solve(G[1..3],explicit):

nops([ab])

3

(5)

evalf([ab]);

[{a = 0., b = 0.}, {a = 0.1542081633e-1, b = 0.2577584543e-2}, {a = -0.1542081633e-1, b = 0.2577584543e-2}]

(6)

eval(G, ab[1]);  #  So,  a=0, b=0, Upsilon=arbitrary

[0, 0, 0, 0, 0, 0, 0, c^2+d^2-1]

(7)

############

Digits:=100;

100

(8)

d1:=fsolve(eval(G[4], ab[2]));

0.1767112612192714286343873125631522838139889970508087652450952127061611264592773044391689682794290833e-1

(9)

fsolve(eval(G[5], ab[2])); %-d1;

0.1767112612192714286343873125631522838139889970508087652450952127047025878451866467897520588192095984e-1

 

-0.14585386140906576494169094602194849e-66

(10)

c1:=fsolve(eval(G[6], [ab[2][], d=d1]));

.9998438534599204826860536133197415830743820837843582966483442787543887342444801865403482145379307231

(11)

fsolve(eval(G[7], [ab[2][], d=d1])); %-c1;

.9998438534599204826860536133197415830743820837843582966483442787543887437298044097515800626952324576

 

0.94853242232112318481573017345e-71

(12)

c1^2+d1^2;

.9999999999999999999999999999999999999999999999999999999999999999999999810143670507891071664311474276

(13)

u1:=arctan(d1,c1);

0.1767204594111244839467484124914988636253117448309960560467697561494599646479420221269303132916932161e-1

(14)

d2:=fsolve(eval(G[4], ab[3]));

-0.1767112612192714286343873125631522838139889970508087652450952127061611264592773044391689682794290833e-1

(15)

fsolve(eval(G[5], ab[3])); %-d2;

-0.1767112612192714286343873125631522838139889970508087652450952127047025878451866467897520588192095984e-1

 

0.14585386140906576494169094602194849e-66

(16)

c2:=fsolve(eval(G[6], [ab[3][], d=d2]));

-.9998438534599204826860536133197415830743820837843582966483442787543887342444801865403482145379307231

(17)

fsolve(eval(G[7], [ab[3][], d=d2])); %-c2;

-.9998438534599204826860536133197415830743820837843582966483442787543887437298044097515800626952324576

 

-0.94853242232112318481573017345e-71

(18)

c2^2+d2^2;

.9999999999999999999999999999999999999999999999999999999999999999999999810143670507891071664311474276

(19)

u2:=arctan(d2,c2);

-3.123920607648680790067968542030352997834638224892006215370267616692870409821414796415341794012947746

(20)

 

Digits:=15:

Conclusion. The solutions of the system are:

[a=0, b=0, Upsilon=any]; [evalf(ab[2])[], Upsilon=evalf(u1)]; [evalf(ab[3])[], Upsilon=evalf(u2)];

[a = 0, b = 0, Upsilon = any]

 

[a = 0.154276352360907e-1, b = 0.257758454297948e-2, Upsilon = 0.176720459411124e-1]

 

[a = -0.154276352360907e-1, b = 0.257758454297948e-2, Upsilon = -3.12392060764868]

(21)

 


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