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Question
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
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Solution
\[\text { Here }, \]
\[f\left( x \right) = 7x - 3\]
\[\text { Let } x_1 , x_2 \text { in R such that } x_1 < x_2 . \text { Then },\]
\[ x_1 < x_2 \]
\[ \Rightarrow 7 x_1 < 7 x_2 \left[ \because 7 >0 \right]\]
\[ \Rightarrow 7 x_1 - 3 < 7 x_2 - 3\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in R\]
\[\text { So,}f\left( x \right)\text { is strictly increasing on R } .\]
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