Question

Write the set of values of a for which f(x) = cos x + a² x + b is strictly increasing on R.

Answer 1

$$\begin{gathered} f\left(x\right)=\cos x+a^{2}x+b \\ f\text{'}\left(x\right)=a^2-\sin x \\ \text{Given:}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) is strictly increasing on}\mathit{\text{R.}} \\ \Rightarrow f\text{'}\left(x\right)>0,\forall x\in R \\ \Rightarrow a^2-\sin x>0,\forall x\in R \\ \Rightarrow a^2>\sin x,\forall x\in R \\ \text{We know that the maximum value of sin}\mathit{\text{x}}\text{is 1.} \\ \text{Since}{a}^2>\sin x,{\mathit{\text{a}}}^2\text{is always greater than 1.} \\ \Rightarrow a^2>1 \\ \Rightarrow {\mathrm{a}}^2-1>0 \\ \Rightarrow \left(a+1\right)\left(a-1\right)>0 \\ \Rightarrow a\in (-\infty ,-1)\cup (1,\infty ) \end{gathered}$$