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Question
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
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Solution
\[f\left( x \right) = x^3 - ax\]
\[f'\left( x \right) = 3 x^2 - a\]
\[\text { Given }:f\left( x \right)\text { is increasing on R }.\]
\[ \Rightarrow f'\left( x \right) \geq 0 \forall x \in R\]
\[ \Rightarrow 3 x^2 - a \geq 0 \forall x \in R\]
\[ \Rightarrow a \leq 3 x^2 \forall x \in R\]
\[\text { The least value of } 3 x^2 \text { is } 0.\]
\[\therefore a \leq 0\]
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