Question about an inequality

Question

I am just wondering, is it true that

$$6 \sqrt{-3 t (1+t)+\sqrt{4+12 t\ \ }\ \ }$$

$$\le\left(\frac{3 t (1+t)+\sqrt{4+12 t}}{1+t+t^2}\ \ \right)^{3/2}+4\sqrt{2}(1-t)^{3/2}$$

for all $\displaystyle t\in[0,1]$?

Thanks!

Answer 1

I did not mess around with the function yet. Do you need a formal proof or what is your purpose for this inequality? A simple Mathematica plot shows that the inequality is most likely true:

Also Mathematica can show your inequality is true:

Reduce[a[t] <= b[t] && 0 <= t <= 1,t]

returns

0 <= t <= 1