@vv  OK, probably you are right. If we give the same initial conditions for a slightly larger t (i took t = 0.1), then a solution exists, in this case f''(0.1) is very large:

l := t -> 0.5*tanh(0.5*t):
deq := diff(f(t), t)*l(t)*(diff(f(t), t, t)*l(t)+9.8*sin(f(t)))+diff(l(t), t)*(diff(f(t), t)^2*l(t)-9.8*cos(f(t))+4*(l(t)-0.5)) = 0, diff(f(t), t, t)=g(t):
sol := dsolve({deq, f(0.1) = 0, D(f)(0.1) = 0.1}, {f(t),g(t)}, numeric);
plots:-odeplot(sol,[t,f(t)],t=0.1..1);
eval(g(t),sol(0.1));
 

@Carl Love   It is better to take the exponent at x of 1.5 instead of 2:

plot(`if`(x>=0 and x<=0.982, 0.015*x^1.5*sin(280*x), NULL), x=0..1.2, -0.02..0.02, color=blue, size=[800,400], gridlines);

I saw two absolutely identical questions for the last 1-2 days and removed the last one as spam. It seems that the second question was automatically deleted. Probably I accidentally pressed the wrong button. I ask you to excuse me for this imprudence.

@John Fredsted 

getMinMax(53^3);

                                                3, 53

Should be   53, 53

See the thread with the same question  http://www.mapleprimes.com/questions/202504-Strange-Behavior-Of--Plotscontourplot

@gtbastos   Try:

restart;
M:=combinat:-permute([0$16,1$16],16):
S:=[seq(`if`(LinearAlgebra:-Modular:-Rank(3, Matrix(4,m, datatype= float[8]))>=3, m, NULL), m=M)]:  # The list of all such matrices, represented as the lists formed of the rows
nops(S);   # The total number of such matrices

@vv  For  x = 0.05 .. 2, y = 0 .. 2  the range of the function is bounded, but rather large. If we increase the lower bound for x-variable, then everything is OK even for the standard grid:

plots:-contourplot(1/(x^2+y^2), x = 0.5 .. 2, y = 0 .. 2, contours = 5, scaling = constrained);

@brian bovril 

seq(max(seq(add(d[s[i]]*x[s[i],s[i+1]], i=1..nops(s)-1) , s=t))<=K, t=Tour2);

d[1]*x[1, 2]+d[2]*x[2, 3]+d[3]*x[3, 4] <= K, max(d[1]*x[1, 2], d[1]*x[1, 3]+d[3]*x[3, 4]) <= K, max(d[1]*x[1, 3], d[1]*x[1, 2]+d[2]*x[2, 4]) <= K, max(d[1]*x[1, 4], d[1]*x[1, 2]+d[2]*x[2, 3]) <= K, max(d[1]*x[1, 2], d[1]*x[1, 3], d[1]*x[1, 4]) <= K

@Earl   Because a straight line intersects a circle at exactly one point if and only if it touches the circle at this point. Of course, for other curves this assertion may be false.

@mattyred99   I have no problems in Maple 2016, Maple 2015, Maple 16 and Maple 12. What version of Maple are you using?  Send a screenshot of your document with this error.

In Maple 2016:

Download Tangents.mw

@Fzen  This is called multiple assignment:

restart;
oldVars:=[a,b,c,d]: newVars:=[1,3,-9,5]:
assign(oldVars=~newVars);

a, b, c, d;
                                          1, 3, -9, 5

Addition. The  myfunc  procedure does the same thing:

myfunc:=proc(x)
local i;
for i from 1 to nops(oldVars) do
if x=oldVars[i] then return newVars[i] fi;
od;
undefined;
end proc:

oldVars:=[a,b,c,d]: newVars:=[1,3,-9,5]:
myfunc~(oldVars);  
                                            [1, 3, -9, 5]

@farjad_etg  It's strange. On my computer (Windows 10)  in Maple 2015.2 and 2016.2  everything works correctly 

The use  eval  and  ~  gives the simplest and shortest solution for the original problem:

sol:=solve([sin(t), 0 < t, t < 8*Pi], t, allsolutions, explicit);
eval~(t, [sol]);

You must provide the objective function so that everyone can try to work with it. And why  xx=0? Are you looking for a minimum of what function?

@mskalsi  I'm not an expert in pictures formats and therefore I cannot answer your question. For the forum, I always use the PNG format (or GIF for animations).