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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!

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    I get $4$. Since you expressed the thing as a function of a single variable t, it's also unclear whether you mean the partial derivative with respect to the first argument or the total derivative wrt t. Either way, I seem to get 4. Are you sure there's no typo? – 2017-01-18
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    well i got 4 when i did it as well but when i entered that as the answer, the program said i was wrong. I then went on to type in 1 and it worked. I don't at all understand why that is the case.. I'm hoping its just an error on the part of my professor. I copied the wording of the question verbatim. – 2017-01-18
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    Yeah, I'm with you there. Either we're both making a similar mistake or they forgot about the $4$ in front of the $t.$ – 2017-01-18

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