.

There's a lot of scope for improving this code below. (A probabilistic fallback might also be worthwhile. I suspect that a fallback that using testeq & signum might not work; testeq is rather limited.)

Note that `is` can depend upon Digits.

> isconvex := proc(expr,A::realcons:=-infinity,B::realcons:=infinity)
> local a,b,t,x,use_expr,res;
>   x := indets(expr,name) minus {constants};
>   if nops(x) = 1 then
>     x := op(x);
>   else error;
>   end if;
>   # For more than one variable, one could test
>   # positive-definiteness of the Hessian.
>   res := is( diff(expr,x,x) >=0 )
>            assuming x>=A, x<=B;
>   if res<>FAIL then
>     return res;
>   else
>     userinfo(1,'isconvex',`falling back to the definition`);
>     # Is it better to expand, or simplify, or put
>     # Normalizer calls around the two-arg evals or
>     # their difference, oder...?
>     use_expr:=expand(expr);
>     is( eval(use_expr,x=(1-t)*a+t*b) <=
>         (1-t)*eval(use_expr,x=a)+t*eval(use_expr,x=b) )
>       assuming a<=b, t>=0, t<=1, b<=B, a>=A;
>   #else
>   ##Is global optimization another fallback?
>   end if;
> end proc:

> # note: this shows that none of the examples
> # below had to fall back to the defn after failing
> # a 2nd deriv test. But many such will exist.
> infolevel[isconvex]:=1:

> isconvex( y^2 );
                                     true

>
> isconvex( y^3 );
                                     false

> isconvex( y^3, 0 );
                                     true

>
> isconvex( (y-3)^2 );
                                     true

>
> isconvex( (y-3)^4 );
                                     true

>
> isconvex( exp(y) );
                                     true

> isconvex( -exp(y) );
                                     false

>
> isconvex( -sin(y), 0, Pi );
                                     true

acer

Thanks for your thinking on this issue. it is very interesting and valuable.

The following is also problematic and the constant function as well:

isconvex(1/(x*x))

Such new property would be also useful in Maple (convex, convace, strictly convex...).
It is pity that Maple doesn't have strong enough basic routines (like is etc.) .
Anyway the isconcave routine is simply to change expr to -expr in isconvex, isnt' it?