Question

Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?

Answer 1

$$f\left(x\right)=x+\cos x+ax+b$$

$$f^{\prime }\left(x\right)=1-\sin x+a$$

$$\text{ For f(x) to be increasing, we must have }$$

$$f^{\prime }\left(x\right)>0$$

$$\Rightarrow 1-\sin x+a>0$$

$$\Rightarrow \sin x<1+a$$

$$\text{ We know that the maximum value of sin x is 1 }.$$

$$\Rightarrow 1+a>1$$

$$\Rightarrow a>0$$

$$\Rightarrow a\in (0,\mathrm{\infty })$$