Describe algorithm to find function maximum

Consider a real-valued, differentiable function f(x) defined on R (or more generally on R^n).

You have access to an oracle that, for any input x, can return both the function value f(x) and its derivative (for n = 1) or gradient (for n > 1), denoted by ∇f(x).

Answer the following:

1. Describe an iterative algorithm to find a (possibly local) maximum of f starting from an initial point x_0.
2. Explain how the sign and magnitude of the derivative or gradient guide the direction of your updates.
3. Discuss how you would choose the step size (learning rate) for each iteration.
4. Describe reasonable stopping criteria and possible issues such as converging to local rather than global maxima or overshooting.

Focus on the high-level algorithm and reasoning; you do not need to provide code.