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A function f is defined by f(x)=|x|^m|x-...

A function `f` is defined by `f(x)=|x|^m|x-1|^nAAx in Rdot` The local maximum value of the function is `(m ,n in N),` `1` (b) `m^nn^m` `(m^m n^n)/((m+n)^(m+n))` (d) `((m n)^(m n))/((m+n)^(m+n))`

A

1

B

0

C

2

D

non of these

Text Solution

Verified by Experts

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`
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