The minimum value of the fuction f(x) given by `f(x)=(x^m)/(m)+(x^(-n))/(n) " where " 1/m+1/n =1" and " m gt 1` is

Options

A. 1
B. 0
C. 2
D. non of these

Answer

A

Explanation

We have ,
`f(x)=x^m/m+x^(-n)/(n)`
`rArr f(x) = x^(m-1)-x^(-n-1) rArr f(x) = (x^(m+n)-1)/(x^(n=1))`
` therefore f(x)=0 `
` rArr x^(m+n)=1 rArr x =1 " " [{:(because m gt 1 and 1/m+1/n = 1 rArr n gt 1 ),( because m+ n gt 2 ):}]`
Now
`f''(x) =(m-1)x^(m-2)+ (n+1)x^(-n-2)`
`rArr f'' (x) = m+ n gt 0 [ because m, n gt 1]`
Hence f(x) is minimum at x=1 . The minimum value of f(x) is given by
` f(1) = 1/m + 1/n = 1`