Prove the inequality ln(x-2)>=2(x-3)/(x-1), x>=3.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll create the helping function f(x) = ln(x-2) - 2(x-3)/(x-1), that is differentiable over the domain of the function.

We'll differentiate and we'll get:

f'(x) = 1/(x-2) - 4/(x-1)^2

f'(x) = [(x-1)^2 - 4x + 8]/(x-2)*(x-1)^2

f'(x) = (x^2 - 2x + 1 - 4x - 8)/(x-2)*(x-1)^2

f'(x) = (x-3)^2/(x-2)*(x-1)^2

We notice that for any value of x, the derivative will be positive, since it's numerator and denominator are always positive.

We also notice that f'(3) = 0

Then the function is increasing over [3 ; +infinite).

For x>=3 => f(x)>=f(3) = 0

If f(x) > 0 => ln(x-2) - 2(x-3)/(x-1)>0

The inequality is true for any value of x>=3: ln(x-2) >=2(x-3)/(x-1).

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