I'm asked to confirm if the CES utility function is convex, and I know it is, I just don't understand why :(
My function is: $$U(x_1,x_2)=(αx_1^ρ+(1-α)x_2^ρ)^{1/ρ}$$ pictured here
I've seen explanations that it's concave because of monotonic transformation and there's always an option to calculate Hessian matrix, which sounds like something I'd make a lot of mistakes in. I just don't understand how do we know that it's a monotonic transformation?
Please please please :(
The CES utility function is not always convex, and nor is it always concave. Indeed, one can show that in your particular case, it will be concave whenever $\alpha$ is in $[0, 1]$ and $\rho \leq 1$.
With regards to your statement about montonic transformations, let us review the following result (see, for example, Jehle and Reny's book on Microeconomic Theory for a proof).
Theorem. Suppose $u: X \to \mathbb{R}$ is a utility function that represents a preference relation $\succeq$ on $X$, and $g: \mathbb{R} \to \mathbb{R}$ is a monotonic function. Then $g \circ u: X \to \mathbb{R}$ is also a utility function that represents $\succeq$.
What you don't seem to grasp is that it doesn't matter whether $u$ itself is monotonic. What the theorem says is that you can always apply a monotonic transformation to a utility function without modifying the preference relation it represents. Another way of saying this is that utility functions are unique up to a monotonic transformation.
In order to show that $u$ is concave for some range of values of $\alpha$ and $\rho$, you may show that $u$ is quasiconcave and homogeneous of degree 1.
Another point of confusion you seem to be having is that preferences and utility functions representing them are two separate objects. The indifference curves associated with the CES function for some range of $\alpha$ and $\rho$ values is convex, but the function itself is quasiconcave when this is the case.
This mimeo on the CES function might come in handy if you're struggling with either step.