Let `f`be a real-valued function satisfying `f(x)+f(x+4)=f(x+2)+f(x+6)`Prove that `int_x^(x+8)f(t)dt`is constant function.

Let `f`be a real-valued function satisfying `f(x)+f(x+4)=f(x+2)+f(x+6)

1.	Let `f`be a real-valued function satisfying `f(x)+f(x+4)=f(x+2)+f(x+6)`Prove that `int_x^(x+8)f(t)dt`is constant function.
Answer» Given that `f(x)+f(x+4)=f(x+2)+f(x+6)`…………1 Replacing `x` by `x+2` we get `f(x+2)+f(x+6)=f(x+4)+f(x+8)`…………………2 From equation 1 and 2 we get `f(x)=f(x+8)`……………3 or `int_(x)^(x+8)f(t)dt+int_(0)^(8)f(t)dt` Thus, `g` is a constant function.

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Given that `lim_(nrarroo) sum_(r=1)^(n) (log_(e)(n^(2)+r^(2))-2log_(e)n)/n = log_(e)2+pi/2-2`, then evaluate : ` lim_(nrarroo) (1)/(n^(2m))[(n^(2)+1^(2))^(m)(n^(2)+2^(2))^(m) "......"(2n^(2))^(m)]^(1//n)`.
`lim_(nrarroo) sum_(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))`
`lim_(nrarroo) (((n+1)(n+2)"........"3n)/(n^(2n)))^(1//n)` is equal to :A. `(27)/(e^(2))`B. `(9)/(e^(2))`C. `3 log3-2`D. `(18)/(e^(4))`