What is the value of 'm' and 'M'?

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Answer 1 · 2017-04-05T13:52:39

$m=1$ and $M=12$

$f'(x)$ is monotonic increasing for $1/2 le x le 1$ and

$f'(1/2)=(192(1/2)^3)/(2+1)=8$

$f'(1)=192/2=96$

then

$f(1/2)+f'(1/2)(x-1/2) le f(x) le f(1/2)+f'(1)(x-1/2)$

but $f(1/2)=0$ so

$int_(1/2)^1f'(1/2)(x-1/2)dx le int_(1/2)^1f(x)dx le int_(1/2)^1f'(1)(x-1/2)dx$

or

$8int_(1/2)^1(x-1/2)dx le int_(1/2)^1f(x)dx le 96int_(1/2)^1 (x-1/2)dx$

or finally

$8/8 le int_(1/2)^1f(x)dx le 96/8$

or

$m = 1$ and $M = 12$