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1.

Prove that the function `f`given by `f(x)=x-[x]`us ubcreasubg ub `(0,1)dot`

Answer» Here, `f(x) = x-[x]`
We know, `x = [x] +{x}`, where `[x]` is the greatest integer function of `x` and `{x}` is the fraction of `x`.
`:. x - [x] = {x}`.
`:. f(x) = {x}`
Let `x_1 = 0.23` and `x_2 = 0.25`
Then, `f(x_2) gt f(x_1)`
It means, if `x_2 gt x_1` , then, `f(x_2) gt f(x_1)` for `x in (0,1).`
Therefore, `f(x)` is an increasing function in `(0,1)`.
Discussion
2.

Show that `f(x)=cos(2x+pi/4)`is an increasing function on `(3pi//8,7pi//8)dot`

Answer» given that `y= f(x)= cos(2x+pi/4)`
`dy/dx> 0 => -2sin(2x+pi/4)> 0`
`2sin(2x+ pi/4) <0`
`sinubrace(2x+ pi/4) < 0 `
`sin a < 0`so, `pi< a<2pi`
so must satisfy
`pi< 2x+pi/4<2pi`
`(3pi)/4 < 2x < (7pi)/4`
`(3pi)/8 < x < (7 pi)/8`
answer
Discussion