Consider the function f(x) = −x 3 + 60x 2 − 900x − 1. Now imagine that the domain of x is the set of real numbers R. Perform 4 iterations of gradient descent algorithm starting from x = 0, with the following settings: step size η = 0.1 step size η = 1

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For this problem, we will need to use the gradient descent algorithm to find the minimum of the function f(x) = −x^3 + 60x^2 − 900x − 1. The gradient descent algorithm works by iteratively updating the value of x based on the negative gradient of the function at that point. The update equation is as follows:

x := x - η * df/dx

where df/dx is the derivative of the function with respect to x and η is the step size.

Let's start by performing 4 iterations of gradient descent starting from x = 0 and with step size η = 0.1.

Iteration 1:
x := 0 - 0.1 * (-3*0^2 + 120*0 - 900) = 90
f(x) = -90^3 + 60*90^2 - 900*90 - 1 = -729001

Iteration 2:
x := 90 - 0.1 * (-3*90^2 + 120*90 - 900) = 54
f(x) = -54^3 + 60*54^2 - 900*54 - 1 = -262189

Iteration 3:
x := 54 - 0.1 * (-3*54^2 + 120*54 - 900) = 44.4
f(x) = -44.4^3 + 60*44.4^2 - 900*44.4 - 1 = -207764

Iteration 4:
x := 44.4 - 0.1 * (-3*44.4^2 + 120*44.4 - 900) = 40.32
f(x) = -40.32^3 + 60*40.32^2 - 900*40.32 - 1 = -188545

Now let's perform 4 iterations of gradient descent starting from x = 0 and with step size η = 1.

Iteration 1:
x := 0 - 1 * (-3*0^2 + 120*0 - 900) = 900
f(x) = -900^3 + 60*900^2 - 900*900 - 1 = -729000001

Iteration 2:
x := 900 - 1 * (-3*900^2 + 120*900 - 900) = -269100
f(x) = -(-269100)^3 + 60*(-269100)^2 - 900*(-269100) - 1 = -2.9628e+24

Iteration 3:
x := -269100 - 1 * (-3*(-269100)^2 + 120*(-269100) - 900) = 8.4441e+10
f(x) = -(8.4441e+10)^3 + 60*(8.4441e+10)^2 - 900*(8.4441e+10) - 1 = -6.5828e+32

Iteration 4:
x := 8.4441e+10 - 1 * (-3*(8.4441e+10)^2 + 120*(8.4441e+10) - 900) = -2.6349e+22
f(x) = -(-2.6349e+22)^3 + 60*(-2.6349e+22)^2 - 900*(-2.6349e+22) - 1 = -1.2977e+69

As we can see from the results, using a larger step size (η = 1) can cause the algorithm to diverge and not find the minimum of the function. In this case, we need to choose a smaller step size (η = 0.1) in order to converge to the minimum.