If $U = f(y+αx) + f(y-αx)$, with $f$ differentiable, show that when $x=0$, then $Ux = 0$.

Question

Can anyone help me with the following execrise?

If $U(x,y) = f(y+αx) + f(y-αx)$, with $f$ differentiable, show that when $x=0$, then $U_x(0,y) = 0$.

Answer 1

Please be more clear in the phrasing of the question and the listing of the assumptions. $U(x,y)=f(y+\alpha x)+f(y-\alpha x)$. Since $f$ is differentiable, $$ U_x(x,y)=\alpha f'(y+ \alpha x)-\alpha f'(y-\alpha x).$$ Plug in $x=0$ to get $U_{x}(0,y)$.