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Question
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
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Solution
\[f\left( x \right) = x^3 - 3 x^2 + 4x\]
\[f'\left( x \right) = 3 x^2 - 6x + 4\]
\[ = 3\left( x^2 - 2x \right) + 4\]
\[ = 3\left( x^2 - 2x + 1 \right) - 3 + 4\]
\[ = 3 \left( x - 1 \right)^2 + 1 > 0, \forall x \in R\]
\[\text { Hence , f(x) is strictly increasing on R } .\]
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