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Question
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
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Solution
\[f\left( x \right) = 3 x^5 + 40 x^3 + 240x\]
\[f'\left( x \right) = 15 x^4 + 120 x^2 + 240\]
\[ = 15 \left( x^4 + 8 x^2 + 16 \right)\]
\[ = 15 \left( x^2 + 4 \right)^2 > 0, \forall x \in R \left[ \because 15 > 0 \text { and } \left( x^2 + 4 \right)^2 > 0 \right]\]
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