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Question
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
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Solution
\[f\left( x \right) = x^3 - 6 x^2 + 12x - 18\]
\[f'\left( x \right) = 3 x^2 - 12x + 12\]
\[ = 3\left( x^2 - 4x + 4 \right)\]
\[ = 3 \left( x - 2 \right)^2 \geq 0, \forall x \text { in R } \left[ \because 3 > 0 \text { &} \left( x - 2 \right)^2 \geq 0 \right]\]
\[\text { So, f(x) is increasing on R } .\]
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