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Prove that the function given by f (x) = x ^3 – 3 x ^2 + 3 x – 100 is increasing in R.

18) Prove that the function given by f (x) = x^3 - 3x^2 + 3x - 100 is increasing in R.

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Given function is,
f (x) = x^3 - 3x^2 + 3x - 100
f^{'}(x) = 3x^2 - 6x + 3
             = 3(x^2 - 2x + 1) = 3(x-1)^2
f^{'}(x) = 3(x-1)^2
We can clearly see that for any value of x in R f^{'}(x) > 0
Hence, f (x) = x^3 - 3x^2 + 3x - 100  is an increasing function in R

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