Conditions for unique, infinite, and no solutions of optimization problem

Question

I have attached the question I am referring to. I assume for the uniqueness one, we can say that the condition is that it has to be a convex function with a global minima. But that does not seem right. And I'm not sure at all about the infinite and no solutions.

Additionally, for part (b) am I to just write the derivative of the function equal to 0?

Hints: What happens if $a$ is positive, negative or zero? If $a$ is zero, what happens if $b$ is positive, negative or zero. (Bonus hint: $c$ does not affect the answers) — Henry, Oct 16 '19 at 23:04
Thank you for that. I wasn't exactly sure what KIND of conditions they were asking for. This confirms it. Can you also take a look at what I asked about the part (b) please? — Aurangzeb Rathore, Oct 16 '19 at 23:13

score 0 · Answer 1 · answered Oct 20 '19 at 16:20

If f''(x) > 0 then the function is concave up(for all x). So it will have minimum. Then it has unique solution if it holds for all value of x.

Now infinitely many solution: In case a = 0 then then f = bx+c. Differentiating it we get f'(x)=b. So it cant be 0 anywhere. So that means it doesn't have a global max or min. This is the case of no solution.

What happens if b=0 then we get f = c. A plane. Now if we differentiate it we have 0 but f''(x) = 0 everywhere. Which means it is an inflection point. But if we look aside in the terrain it is always the same. So it has infinitely many minimas.

@Henry.: Can you check the answer reasoning? – omega Oct 20 '19 at 16:21 — omega, Oct 20 '19 at 16:21

Conditions for unique, infinite, and no solutions of optimization problem

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