Well, after looking further in that book, I found some very cool techniques to solve this, and it involves algebraic inequalities; in this problem, the Cauchy-Bunhiacowsky-Schwartz inequality, or CBS for short(edit: and the AM-GM inequality). It will take me some time to try and solve the problem using that technique.

Turns out that when a = b = c =1 then each radical term evaluates to sqrt(5). So now we are left with:

Does a = b = c =1 minimize 3a^2 + 2/(b+1) + c^4? I don't think it does. If you replace b+1 with 4-a-c we now get:

f(a,c) = 3a^2 + 2/(4-a-c) + c^4.

Let pf/pa represent the "partial of f wrt a" - I don't have a lower case delta on my keyboard.

I then found pf/pa and pf/pc and set them = 0 but a = c = 1 does not produce a 0 value for the partials. I'm stuck.

Due to the symmetry I think we can this reduce to:

Show that the minimum value of 3a^2 + 2/(b+1) + c^4 is 5 given the constraint that

a + b + c = 3. I'll try this and see where it leads.