Prove that the function given by $f (x) = x^{3} - 3 x^{2} + 3 x - 100$ is inreasing in R.

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Solution

$f (x) = x^{3} - 3 x^{2} + 3 x - 100$
On differentiating both sides w.r.t. x, we get,
$f^{'} (x) = 3 x^{2} - 6 x + 3 = 3 (x^{2} - 2 x + 1) = 3 (x - 1)^{2} \geq 0$ for real values of x.
Therefore, f is increasing in R.

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