Find range of $a$ if $f(x)$ is increasing $\forall $ $x \in \mathbb{R}$

Question

Find range of $a$ if $f(x)$ is increasing $\forall $ $x \in \mathbb{R}$ where

$$f(x)=\int_{0}^{x} \left|\log_2\left(\log _3\left(\log_4\left(\cos t+a\right)\right)\right)\right|dt$$ where $|.|$ is absolute value function

My Try:

we have

$$f'(x)=\left|\log_2\left(\log _3\left(\log_4\left(\cos x+a\right)\right)\right)\right| \gt 0$$

any clue to find range of $a$?

Answer 1

The value of the derivative can not be negative since it is an absolute value, however it can be zero depending on the value of a, then we must calculate the values of a that can make zero the derivative, to do it proceed as follows:

|log2(log3(log4(cosx+a)))|=0
log2(log3(log4(cosx+a)))=0
log3(log4(cosx+a))=1
log4(cosx+a)=3
cosx+a=(4^3)
cosx+a=64
a=64-cos(x) (Remember that cos (x) ∈ [-1, 1] )
if a ∈ [63,65] then |log2(log3(log4(cosx+a)))| Will be zero for an infinity of values of x. 

Then if "a" takes values between [63, 65] the derivative will be zero for an infinity of values of x. Then this range of values must be excluded, so in order that the derivative is always positive, ie greater than zero, a ∈ (-infinite, 63) ∪ (65, infinity), however in order that cos (x) + a is always positive "a" must be greater than 1, so the complete solution is:

 a ∈ (1, 63) ∪ (65, infinity).