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Question
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
Solution
\[f\left( x \right) = \cos x + a^2 x + b\]
\[f'\left( x \right) = a^2 - \sin x\]
\[\text { Given :f(x) is strictly increasing on R }.\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in R\]
\[ \Rightarrow a^2 - \sin x > 0, \forall x \in R\]
\[ \Rightarrow a^2 > \sin x, \forall x \in R\]
\[\text { We know that the maximum value of sin x is 1 }.\]
\[\text { Since } a^2 > \sin x, a^2\text { is always greater than 1 }.\]
\[ \Rightarrow a^2 > 1\]
\[ \Rightarrow a^2 - 1 > 0\]
\[ \Rightarrow \left( a + 1 \right)\left( a - 1 \right) > 0\]
\[ \Rightarrow a \in ( - \infty , - 1) \cup (1, \infty )\]
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