Question:
Prove that the function given by $f(x)=x^{3}-3 x^{2}+3 x-100$ is increasing in $\mathbf{R}$.
Solution:
We have,
$\begin{aligned} f(x) &=x^{3}-3 x^{2}+3 x-100 \\ f^{\prime}(x) &=3 x^{2}-6 x+3 \\ &=3\left(x^{2}-2 x+1\right) \\ &=3(x-1)^{2} \end{aligned}$
For any $x \in \mathbf{R},(x-1)^{2}>0$
Thus, $f^{\prime}(x)$ is always positive in $\mathbf{R}$.
Hence, the given function (f) is increasing in R.
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