$U(c_t)=\sum_{t=0}^{\infty}\beta^t(\{u_0c_t+\frac{u_1}{2}c_t^2\})$ subject to $c_t+k_{t+1}\leq f_0 k_t$

I need to find the euler equations and the transversality conditions.

I have currently tried using the Bellman method to find the euler equations, which gave me the following: $U'(c_t)=\beta(U'(c_{t+1})f_0)$, but I do not think is correct. How would I get to the transversality conditions from here? Can you check my Euler equation?



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