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Question
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Solution
\[f\left( x \right) = x^3 - 15 x^2 + 75x - 50\]
\[f'\left( x \right) = 3 x^2 - 30x + 75\]
\[ = 3 \left( x^2 - 10x + 25 \right)\]
\[ = 3 \left( x - 5 \right)^2 > 0, \forall x \in R \left[ \because \text { Square of any function is always greater than zero } \right]\]
\[\text{ So,f(x)is an increasing function for all x} \in R.\]
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