Question

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

Answer 1

`m=1` and `M=12`

Explanation:

`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`