Question

Is `f(x)= cos(x+(5pi)/4)` increasing or decreasing at `x=-pi/4`?

Answer 1

See explanation.

Explanation:

Generally if the function `f(x)` has the derrivative `f^'(x_0)` then we can say that:

• `f(x)` is increasing at `x_0` if `f^'(x_0)>0`

• `f(x)` is decreasing at `x_0` if `f^'(x_0)<0`

• `f^'(x)` may have an extremum at `x_0` if `f^'(x_0)=0` (additional test is required)

In the given example we have:

`f^'(x)=-sin(x+(5pi)/4)`

`f^'(x_0)=-sin(-pi/4+(5pi)/4)=-sin(pi)=0`

`f^'(x_0)=0`, so `f(x)` has either an extremum, or an inflection point.
To check if the point is extremum we have to check if the first derivative changes sign at `x_0`.

graph{(y+sin(x+(5pi)/4))((x+pi/4)^2+(y^2)-0.01)=0 [-4, 4, -2, 2]}

At `x_0=-pi/4` the derrivative changes sign from negative to positive, so the point is a minimum.