Question

Is `f(x)= sin(x+(3pi)/8)` increasing or decreasing at `x=pi/12`?

Answer 1

The function `f` is increasing at point `x = pi/12`.

Explanation:

The growth or decrease of a function is given by the sign of its first derivative. If the derivative is positive, the function is increasing, whereas if the derivative is negative, the function will be decreasing.

Therefore, we first calculate the derivative of `f (x)`:

`f' (x) = cos (x + {3 pi}/8)` .

and then we substitute the value of `x` for `pi / 12` to calculate the sign of the derivative at that point:

`f' (pi/12) = cos (pi/12 + {3 pi}/8) = cos ({11 pi}/24)`

Given that:

`0 lt {11 pi}/24 lt pi/2`

then

`1 gt cos ({11 pi}/24) gt 0`

and, therefore, the function is increasing at `x = pi / 12`.