We have been finding a set of optimal parameters for the linear least-squares regression minimization problem by identifying critical points, i. e. points at which the gradient of a function is the zero vector, of the following function: F(αo, ... ,αm) = ΣNn = 1(α0, + α1xn1 + ... + αMxnM - yn)2
Please help to justify this methodology in the following way. Letting G: Rd - R be any function that is differentiable everywhere, show that, if G has a local minimum at a point xo, then its gradient is the zero vector there, i. e., ΔG(x0) = 0.

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step-by-step explanation:

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

12.5 feet

step-by-step explanation: