if I have to maximize $\left[\int_0^N c(i)^k\,\mathrm di\right]^{1/k}$ subject to $\int_0^N p(i) c(i) \,\mathrm di \leq I$ where $I, k$ are constants, and $c(i)$ is our choice variable.
I saw in the solution that the FOC of this problem is: $k c(i)^{k-1} = \lambda p(i)$.
Can anyone please help explain why it's the case that we can simply differentiate the integrand?
Thank you.
