If `f(x)=alphax^(n)`, prove that `alpha=(f^(')(1))/n`.

If f(x)=alpha x^(n), prove that alpha=(f'(1))/(n)

If g is the inverse of f and f'(x)=(1)/(1+x^(n)) prove that g'(x)=1+(g(x))^(n)

If f(a+x)=f(x), then prove that int_(a)^(na)f(x)dx=(n-1)int_(0)^(a)f(x)dx where a>0 and n in N.

If f(x)=(x-alpha)^(n)g(x) and f(alpha)=f'(alpha)=f''(alpha)=f^(n-1)(alpha)=0 where f(x) and g(x) are polynomials.For polynomial f(x) and g(x) with rational cofficients,then answer the following quily one (1) If y=f(x) touches the x -axis at only one point,then the point of contact

If f(x)=x^(n),"n" epsilon N , then the value of f(1)-(f^(')(1))/(1!)+(f^(")(1))/(2!)-(f^''')(1)/(3!)+…+(-1)^(n)(f^(n)(1))/(n!) is

if f(x)=x^(n) then the value of f(1)-(f'(1))/(1!)+(f''(1))/(2!)+--+((-1)^(n)f^(prime--n)xx(1))/(n!)