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Question: A function f is defined by f(x) = \| x \|<sup>m</sup> \|x – 1\|<sup>n</sup>" x Ī R. The maximum valu...

A function f is defined by f(x) = | x |m |x – 1|n" x Ī R. The maximum value of the function is (m, n Ī N) –

A

1

B

mnnm

C
D

) (mn)mn(m+n)m+n\frac { ( \mathrm { mn } ) ^ { \mathrm { mn } } } { ( \mathrm { m } + \mathrm { n } ) ^ { \mathrm { m } + \mathrm { n } } }

Answer
Explanation

Solution

f(x) = {(−1)m+nxn(x−1)n if x<0(−1)nxm(x−1)n if 0≤x<1xm(x−1)n if x≥1\left\{ \begin{array} { c c } ( - 1 ) ^ { m + n } x ^ { n } ( x - 1 ) ^ { n } & \text { if } x < 0 \\ ( - 1 ) ^ { n } x ^ { m } ( x - 1 ) ^ { n } & \text { if } 0 \leq x < 1 \\ x ^ { m } ( x - 1 ) ^ { n } & \text { if } x \geq 1 \end{array} \right.

g(x) = xm (x – 1)n, then

g '(x) = mxm – 1 (x –1)n + nxm (x – 1)n –1

= xm –1 (x –1)n –1 {mx – m + nx} = 0

f '(x) = 0 Ž g '(x) = 0 Ž x = 0, 1 or mm+n\frac { \mathrm { m } } { \mathrm { m } + \mathrm { n } }

f(0) = 0, f(1) = 1 and

f = (–1)n mmnn(−1)n(m+n)m+n\frac { \mathrm { m } ^ { \mathrm { m } } \mathrm { n } ^ { \mathrm { n } } ( - 1 ) ^ { \mathrm { n } } } { ( \mathrm { m } + \mathrm { n } ) ^ { \mathrm { m } + \mathrm { n } } }

Ž mmnn(m+n)m+n\frac { m ^ { m } n ^ { n } } { ( m + n ) ^ { m + n } } > 1 [0<mm+n<1]\left[ 0 < \frac { \mathrm { m } } { \mathrm { m } + \mathrm { n } } < 1 \right]

The maximum value = mmnn(m+n)m+n\frac { m ^ { m } n ^ { n } } { ( m + n ) ^ { m + n } }