I was given the following problem in an assignment:
Consider the function $$f(t,x,y)=3\cos(x-t)+2xy+(8+4t)\cos(y)$$
and note that $(x,y)\mapsto f(0,x,y)$ has a local maximum at $(x,y)=(0,0)$. As $t$ varies, the function $(x,y)\mapsto f(t,x,y)$ at the nearby point $(x,y)=(\xi(t),\eta(t))$ in such a way that $(\xi(0),\eta(0))=(0,0)$ and $t\mapsto (\xi(t),\eta(t))$ is a differentiable function of t.
What is $\frac{\partial}{\partial t}f(t,\xi(t),\eta(t))$ when $t=0$?
It turns out that the answer is $1$ but I do not understand why. Would someone be able to walk me through the steps of this problem?
Thanks!