Question

A function $f$ is defined by $f\left(x\right)=|x{|}^m|x-1{|}^m,\forall x\in R$.
The maximum value of the function is ? $(m,n\in N)$

• A. $1$
• B. $m^n{n}^m$
• C. $\frac{m^m{n}^n}{(m+n{)}^{m+n}}$
• D. $\frac{(mn{)}^{mn}}{(m+n{)}^{m+n}}$

Answer 1

Answer: (C)

$\frac{m^m{n}^n}{(m+n{)}^{m+n}}$

$f\left(x\right)={\left|x\right|}^m{|x-1|}^m$

$f^{\prime }\left(x\right)=m{x}^m{(x-1)}^{m-1}+m{x}^{m-1}{(x-1)}^m$

Now for minima/maxima,$f^{\prime }\left(x\right)=0$

$m{x}^m{(x-1)}^{m-1}=-m{x}^{m-1}{(x-1)}^m$

$x=-(x-1)$

$x=\frac{1}{2}$

Now, $f\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^m{(\frac{1}{2}-1)}^m$

$f\left(\frac{1}{2}\right)=\frac{\left(1\right)}{{\left(2\right)}^{m+m}}$

Comparing with options

$f_{max}=\frac{m^m{n}^n}{{(m+n)}^{m+n}}\Rightarrow \left[\frac{1^m\times 1^n}{{(1+1)}^{m+n}}\right]$ Answer