Variation Of Graphs, Periodicity, and Extreme values

Variation Of Graphs, Periodicity, and Extreme values
Definition (periodic function, period)
    Let \(A \subseteq R\) and \(f:A\to R\) be a function .  Then ‘f’ is called “periodic function” if there exists a positive real number ‘p’ such that
i)  \(x + p \in A\)   for all \(x \in A\) and
ii)   \(f(x + p) = f(x)\)  for all \(x \in A\)
    if such a positive real number exists then it is called a period of ‘f’
For example : \(f(x) = x - [x]\) for all \(x \in R\)
        where [x] = Integral part of x
        \(f(1 + x) = (1 + x) - [1 + x]\)
        \(= 1 + x - \{ 1 + [x]\} (\because [1 + x] = 1 + [x])\)
        \(= 1 + x - 1 - [x]\)
        \(= x - [x] = f(x)\)
 i.e.,  \(f(x + 1)\,\,\, = f(x)\,\,\,\,\,\,\,\,\,\,[\because f(x + p) = f(x)],\,\,\,\,\,p = 1\)
    ‘1’ is called period of ‘f’
Note :- f(x)=K for all \(x \in R\) , any positive real number is period of ‘f’.
Periods of Trigonometric functions
    For any real number \(\theta\) , we have observed that  \(\theta\) and \(2\pi+\theta\) have same trigonometric ratios.
    f(x)=sinx
    \(f(2\pi + x) = \sin (2\pi + x) = \sin x\)

    \(f(4\pi + x) = \sin (4\pi + x) = \sin x\)
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    \(f(n\pi + x) = \sin (n\pi + x) = \sin x\)
    Here ‘n’ is even multiple of ‘\(\pi\)‘
    Here the least value of ‘p’ is “\(2\pi\)”  among \(2\pi ,4\pi ,6\pi ...........n\pi \)
    period of f(x)=sinx  is \(2\pi\)
   Similarly, period of cosx is \(2\pi\)
    period of \(\tan x\,\,is\,\,\,\,\pi \)
Note : Period of reciprocal functions of the above  and are i.e., cosx, secx, and cotx respectively are \(2\pi ,2\pi \) and \(\pi\)
Note 2:- Let \(f:R \to R\) be a periodic function and ‘p’ be a period of ‘f’.  Let a, b, c be real constants such that \((a \ne 0)\) then the function \(g:R \to R\)  defined by \(g(x) = f(ax + b) + \) \(c\rlap{--} Vx \in R\)also periodic \(\frac{p}{{|a|}}\) and is the period of ‘g’ further, if ‘p’ is period of f then \(\frac{p}{{|a|}}\) is the period of ‘g’.
For example : \(f(x) = \sin (5x + 3)\) for all \(x\in R\)
Solution :    We know that the function \(g(x) = \sin x\,\rlap{--} V\,x \in R\) has the period \(2\pi\) , now\(f(x) = g(5x + 3)\)      
     period of \(f(x) = \frac{p}{{|a|}} = \frac{{2\pi }}{{|5|}} = \frac{{2\pi }}{5}\)
Example 2 :- Period of \(f(x) = \sin (5x + 6) + 11\)
Solution :-    Period of \(\sin x = 2\pi \)
            \(f(x) = \sin (5x + 6) + 11\)
             period of \(f(x) = \frac{p}{{|a|}} = \frac{{2\pi }}{{|5|}} = \frac{{2\pi }}{5}\)
Example 3:- Period of \(f(x) = a.\sin (2x + 3) + b\)
            period = \(\frac{p}{{|a|}} = \frac{{2\pi }}{{|2|}} = \frac{{2\pi }}{2} = \pi \)

Note 1:-If f₁(x) and  f₂(x)  are periodic functions with periods p₁ and p₂  respectively then the function \(a.{f_1}(x) + b.{f_2}(x)(a,b \in R)\) is also periodic and its period is p which is the L.C.M of  p₁ and p₂
Example:-  
    period of \(f(x) = a\tan x + b.\cos x\)
    period of  \(\tan x\,\,is\,\,\pi \)
    L.C.M of \(\pi ,\,\,\,2\pi \,\,is\,\,2\pi \)
      The period of \(f(x) = a\tan x + b.\cos x\)  is  \(2\pi\)
Note 2:- If \({f_1}(x),{f_2}(x),{f_3}(x)\)  and \({f_4}(x)\) are periodic functions with periods  \({p_1},{p_2},{p_3}\)and p₄  respectively, then the period of \(\frac{{a.{f_1}(x) + b.{f_2}(x)}}{{c.{f_3}(x) + d.{f_4}(x)}}\)  where \(a,b,c,d \in R\)  is p which is the L.C.M of  \({p_1},{p_2},{p_3}\& {p_4}\).
Ex : Find the period of \(f(x) = \frac{{5\sin x + \cot x}}{{11\cos x + 3\tan x}}\)

period of  \(\sin x\, = \,2\pi \to {p_1}\,\)
period of \(\cot x\, = \,\pi \to {p_2}\)
period of \(\cos x\, = \,2\pi \to {p_3}\)
 period of \(\tan x\, = \,\pi \to {p_4}\)
L.C.M of \({p_1},{p_2},{p_3},{p_4}\,\, = \,\,2\pi \)