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The images of 1: y = x 3 and y = 5x+2 have a unique intersection in the first quadrant, that is, the equation x 3 = 5x+2 has only one positive root.

2: Let f (x) = x 3-5x-2, then f (2) =-4.

So the real root of f (x) = x 3-5x-2 = 0 must be in the interval (2,3).

Divide (2,3) by (2,2.5) u (2.5,3)

And because f (2.5) =1.125 >; 0

So the real root of f (x) = x 3-5x-2 = 0 must be in the interval (2,2.5).

Divide (2,2.5) by (2,2.25) u (2.25)U(2.25)

Because f (2.25) =- 1.859.

So the real root of f (x) = x 3-5x-2 = 0 must be in the interval (2.22, 2.5).

Divide (2.25, 2.5) by (2.375, 2.5)U(2.25, 2.5)

Because f (2.375) =-0.4785.

So the real root of f (x) = x 3-5x-2 = 0 must be in the interval (2.375, 2.5).

Divide (2.375, 2.5) into (2.375, 2.4375)U(2.4375, 2.5)

And because f (2.4375) = 0.2946 >: 0.

So the real root of f (x) = x 3-5x-2 = 0 must be in the interval (2.3750, 2.4375). If we take any value in this interval, such as x0 = 2.3892, the error between it and the real root of x 3-5x-2 = 0 is less than 0.00 1.

Therefore, any value of the interval (2.3750, 2.4375) satisfies the condition.