For a homework question, I am given the following:
Let $f: \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x,y) = \alpha x^2 + \beta xy+ \gamma y^2$ for some $\alpha , \beta , \gamma \in \mathbb{R}^2$.
(A) Compute $Df(x)$ for $x \in \mathbb{R}^2$
(B) Show $Df(x)(y) = Df(y)(x)$ for all $x,y \in \mathbb{R}^2$
(C) Show $Df(x)(x) = 2f(x)$
For part (A), I just took the partial derivative of f with respect to x, but for parts (B) and (C) I am running into problems with how to interpret the question. Here are my assumptions:
For B, I am assuming this is asking for us to show that the partial derivative of x and then y is equal to the partial derivative of y and then x.
For C, I am assuming this is asking us to take the partial derivative with respect to x twice
How would you all interpret parts (B) and (C)?