If `f(x)=(a^x)/(a^x+sqrt(a ,)),(a >0),` then find the value of g(n)=`sum_(r=1)^(2n-1)2f(r/(2n))` g(4)

Find the sum sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1)) .

Iff(x)=e^(g(x))a n dg(x)=int_2^x(tdt)/(1+t^4), then find the value of f^(prime)(2)

If f(x) is integrable over [1,], then int_1^2f(x)dx is equal to ("lim")_(nvecoo)1/nsum_(r=1)^nf(r/n) ("lim")_(nvecoo)1/nsum_(r=n+1)^(2n)f(r/n) ("lim")_(nvecoo)1/nsum_(r=1)^nf((r+n)/n) ("lim")_(nvecoo)1/nsum_(r=1)^(2n)f(r/n)

If ("lim")_(xveca)[f(x)+g(x)]=2a n d ("lim")_(xveca)[f(x)-g(x)]=1, then find the value of ("lim")_(xveca)f(x)g(x)dot

if = |{:(x^(n),,n!,,2),(cos x,,cos.(npi)/(2),,4),(sin x,,sin .(npi)/(2),,8):}|, then find the value of (d^(n))/(dx^(n)) [f(x)]_(x=0) .(n in z) .

if =|{:(x^(n),,n!,,2),(cos x,,cos.(npi)/(2),,4),(sin x,,sin .(npi)/(2),,8):}|, then find the value of (d^(n))/(dx^(n)) [f(x)]_(x=0) .(n in z).

Statement1: if n in Na n dn is not a multiple of 3 and (1+x+x^2)^n=sum_(r=0)^(2n)a_r x^r , then the value of sum_(r=0)^n(-1)^r a r^n C_r is zero Statement 2: The coefficient of x^n in the expansion of (1-x^3)^n is zero, if n=3k+1orn=3k+2.

Let f(x)=int_0^x(dt)/(sqrt(1+t^3))a n dg(x) be the inverse of f(x) . Then the value of 4(g^(primeprime)(x))/((g(x))^2)i s____

If f(x) and g(x)=f(x)sqrt(1-2(f(x))^2) are strictly increasing AAx in R , then find the values of f(x)dot

If f(x)={1-|x|,|x|lt=1 0,|x|>1'a n dg(x)=f(x-1)+f(x+1), find the value of int_(-3)^3g(x)dxdot