wolfram alpha
Hopefully, this kind of question is ok, have seen a couple of other WA-queries that hasn't been downvoted. Apologies if not.
I want to minimize say $f(x)=ax^2+bx+c$. Wolfram Alpha gives me an answer for different intervals of $a$ which makes sense. However, Is there a way to explicitly specify $a>0, b>0, c>0$?
Optimization problems generally assume real input an output. Accordingly, WolframAlpha assumes this type of input. You can get around this issue by recasting your problem in terms of real real variables. For example:
minimize |4 (x + I*y)^2 + 1| for x^2+y^2==1
or
minimize |4*exp(I*t) + 1|
Whether or not things are "undefined" largely depeneds on what framework you are working in.
If we are working in the naturals, we might say that $3-5$ is undefined.
There are many systems where it makes sense to assign $\frac{n}{0}$ some value. In this particular example, it is defined to be complex infinity, which can be thought of as follows: suppose we are looking at the complex plane. Similarly to how the complex number "0" is represented by a zero vector of arbitrary direction, we wish to associate all complex numbers of infinite absolute value (regardless of direction) to a single point.
This is complex infinity, and geometrically, by associating all complex numbers of infinite absolute value to be the same on the plane, we have formed a sphere, one with zero on the bottom, and complex infinity on the top. This is called the Riemann Sphere.
Best Answer
Yes, you can simply include those constraints: minimize ax^2+bx+c, a>0, b>0, c>0
Alternatively, you can complete the square and observe the extreme value occurs when $x=-\frac b{2a}$: complete the square ax^2+bx+c