this is a concept about rational and irrational numbers.

we know that cosx is a periodic function.

so we can say that an equation like       cos2x + cos4x          will attain maximum value at many places, whenever cos2x and cos4x are BOTH = 1 we will have maximum value.

example, if we say that x = 0 gives the value =2 for the above eqn.. we see that x = 2 also gives eqn = 2. so 2 is ONE OF THE PERIODS  of this eqn. (not the fundamental period, which is /2 )

this only happens because 2 and 4 are BOTH RATIONAL numbers, so they have a LCM and HCF.

but in your question, we have one rational numb and 1 irrational numb.. they do NOT have any LCM, so that means that the equation   cosx + cos 2^1/2x     is NON PERIODIC .

this means that the function will take its maximum value for ONLY ONE value of x.

by inspection we see that x = 0 is this value.

hope u get it.