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

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

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

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

Let f(x) =(alphax)/(x+1) Then the value of alpha for which f(f(x) = x is

If f(x)=(x-alpha)^ng(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 questions (1)If y = f(x) touches the x-axis at only one point, then the point of contact

If f(x)=(x-alpha)^ng(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 questions (1)If y = f(x) touches the x-axis at only one point, then the point of contact

If f(x)=(x-alpha)^ng(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 questions (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+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.

If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.