(1) The function can be defined separately where

\sin x

takes positive values between

2 k π

and

(2 k + 1) π

(for integer

k

), that is in the first two quadrants, and where

\sin x

takes negative values between

(2 k + 1) π

and

(2 k + 2) π

, that is in the third and fourth quadrants.

\begin{matrix} f (x) & = 2 \sin x for 2 k π \leq x \leq (2 k + 1) π \\ = 0 for (2 k + 1) π \leq x \leq (2 k + 2) π . \end{matrix}

The first derivative:

\begin{matrix} f' (x) & = 2 \cos x for 2 k π < x < (2 k + 1) π \\ = 0 for (2 k + 1) π < x < (2 k + 2) π \end{matrix}

and the derivative

f' (x)

is not defined at the points

x = k π

for any integer

k

(2)

\begin{matrix} f (x) & = \sin x + \cos x = A \sin (x + α) \\ = A \sin x \cos α + A \cos x \sin α \end{matrix}

implies that

A \sin α = 1 and A \cos α = 1

and hence

\tan α = 1

α = π / 4

and

A = \sqrt{2}

. Hence the graph of this function is a sine graph with a phase shift of

π / 4

, that is

f (x) = 0

when

x = k π - π / 4

, it takes the value 1 when

x = 2 k π

and

x = 2 k π + π / 2

and the value -1 when

x = (2 k + 1) π

and

x = 2 k π - π / 2

, has maximum values

(2 k π + π / 4, \sqrt{2})

and minimum values

((2 k + 1) π + π / 4, - \sqrt{2})

\begin{matrix} f (x) & = \sin x - \cos x = B \sin (x + β) \\ = B \sin x \cos β + B \cos x \sin β \end{matrix}