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))`

Options

A. 1
B. `m^(n)n^(m)`
C. `(m^(m)n^(n))/((m+n)^(m+n))`
D. `(mn^(mn))/(m+n^(m+n))`

Answer

3

Explanation

We have f(X) =`{{:((-1)^(m+n)x^(m)(x-1)^(n),if xlt0),((-1)^(n)x^(m)(x-1)^(n),if 0lexlt1),(x^(m)(x-1)^(n),if xge1):}`
Let `g(X) =x^(m)(x-1)^(n)` Then
`g(x) = mx^(m-1)(x-1)^(n)+nx^(m)(x-1)^(n-1)`
`=x^(m-1)(x-1)^(n-1){mx-m+nx}`
Now `f(X) =0 or g(x)=0ni.e x=0 ,1 or (m)/(m+n)`
f(X)=0 f(1)=0 and
`f((m)/(m+n))=(-1)^(n)m^(mn^(n)(-1)^(n))/(m+n)^(m+n)=(m^(m)n^(n))/(m+n)^(m+n)gt0`