Solving Equation: Find t_1,t_2 for C^1 Fcn of t_1,t_2

[Benny]

Hi, I'm stuck on the following implicit function question.

Q. Find the values of t_1, t_2 such that every solution theta is determined as a C^1 function of t_1 and t_2.

The equation is \theta ^3 + t_1 \theta + t_2 = 0.

Ok from what I gather we want \theta = g\left( {t_1 ,t_2 } \right). This one is a little different to the others I've done before and I think one of the conditions I need to check, for this particular case is:

<br /> \frac{{\partial F}}{{\partial \theta }} = 3\theta ^2 + t_1 \ne 0,F\left( {t_1 ,t_2 ,\theta } \right) = \theta ^3 + t_1 \theta + t_2 <br /> where the not equal to zero condition is satisified at a specific point (..,..,..).

Of course if I want to apply the implicit function theorem I need a specific point (t_1,t_2), maybe call it (x,y) to avoid ambiguity. Usually I'm given
one but in this case I'm not so I'm at a loss as to what I need to do. I was told that this question requires a bit of thought but I'm too stupid to work this out so can someone help me out? Any help would be good thanks.

 

[Benny]

Using the implicit function theorem it came down to \frac{{\partial F}}{{\partial \theta }} = 3\theta ^2 + t_1 \ne 0.

But like I said before I'm not given a specific point to work with as is usually the case. The above is what I obtained and the answer is that C^1 solutions exist when 4t_1 ^3 + 27t_2 ^2 \ne 0. I cannot see how they came to that conclusion and I don't understand how you could go any further than what I have given as my answer. Can someone please help me out?

 

[vaishakh]

Just think about what a-zero equation represent? what does a zero-equation physically mean? Since you have simply asked this in Maths column, I am asking you to consider this in a co-ordinate system. Now what does their differentiation of that mean, like z = f(x, y).

 

[Benny]

To be honest I really cannot figure out what you appear to be suggesting. Just to make things clear, there was no differentiation, or any other operation for that matter, that was given in the stem of the question. I decided to carry out the differentiation as it is a requirement in the theorem which I think is supposed to be used.

I do not view this particular question as anymore than the application of a theorem. I firmly believe that I am simply not seeing something very simple. This problem doesn't even look all that complicated. But I cannot even start because I can't make any connections.