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Question
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
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Solution
\[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\]
\[f'\left( x \right) = 12 x^2 - 36x + 27\]
\[ \Rightarrow f'\left( x \right) = 3\left( 4 x^2 - 12x + 9 \right)\]
\[ \Rightarrow f'\left( x \right) = 3 \left( 2x - 3 \right)^2 > 0, \forall x \in R\]
So, f(x) is increasing on R.
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