The question I am posting today relates to convexity analysis of a function. Specifically, proving that a function is convex in $\mathbb{R}^2$ definition. Namely, that a function f, is convex if for all $t \in [0,1]$ and $a,b \in \mathbb{R}^2$, if $f((1-t)a+tb)\leq (1-t)f(a)+tf(b)$.
I have the following function: \begin{equation} x^2+2xy+y^2+\beta x+\beta y \end{equation} where $\beta$ is a parameter value.
Now, I've done the leg work (correctly I believe) to simplify the two components of the definition properly.
Namely, if $f((1-t)a+tb)$ is (1) and $(1-t)f(a)+tf(b)$ is (2) then I've got the following equations: (1) \begin{equation} =\beta((1-t)(a_1+a_2)+t(b_1+b_2))+(1-t)^2(a_1^2+a_2^2)+t^2(b_1^2+b_2^2)+ \\(2t-2t^2)(a_1b_1+a_2b_2) \end{equation}
(2) \begin{equation} =\beta((1-t)(a_1+a_2)+t(b_1+b_2))+(1-t)(a_1^2+a_2^2)+t(b_1^2+b_2^2)+\\ 2((1-t)(a_1a_2)+t(b_1b_2)) \end{equation}
Now, in terms of proving that (1) $\leq$ (2), I think my strategy has been corect. I've shown that the first term of (1) and (2) are the same, and that then second term of (1) $\leq$ (2) because the $(1-t)^2$ and (t^2) mean that for t $\in$ (0,1) that number will be less than (1-t) and t respectively.
It's also true that the second terms of (1) and (2) are equal when t=0 or t=1.
However my problem is the third term. I can't reduce that third term down such that \begin{equation}(2t-2t^2)(a_1b_1+a_2b_2)\leq2((1-t)(a_1a_2)+t(b_1b_2))\end{equation}
This is a problem because I'm sure it means I haven't proven (1)$\leq$(2). I wrote a program in matlab to check if that third term would in practice be problematic and it was for $(a_1,b_1)<0$ or $(a_2,b_2)<0$. Any exceptions defy the definition of convexity so there's got to be some mistake that I've made but I can't seem to find it.
I would really appreciate some advice about moving forward here in terms of the proof. Either there's some mistake in my algebra, or in the way I've interpreted how to do this proof or some next step I can use to show that (1)$\leq$(2).
Thank you all in advance.
Note: I have tried to show the convexity with a Hessian matrix but I'm not sure that's allowed in terms of the specific requirement to show convexity in $\mathbb{R}^2$