Question

Is `f(x)= cos(x+(pi)/4)` increasing or decreasing at `x=pi/3`?

Answer 1

`f(x)=cos(x+pi/4)` is decreasing at `x=pi/3`

Explanation:

Whether a function is increasing or decreasing at a point depends on the value of its derivative at that point.

If derivative is positive, function is increasing and if it is negative, function is decreasing.

As `f(x)=cos(x+pi/4)`, `f'(x)=-sin(x+pi/4)`.

and at `x=pi/3`, `f'(pi/3)=-sin(pi/3+pi/4)=-sin((7pi)/12)=-sin(pi-(5pi)/12)=-sin((5pi)/12)`.

As `sin((5pi)/12)>0` (`(5pi)/12` being acute angle), `f'(pi/3)<0`

Hence, `f(x)=cos(x+pi/4)` is decreasing at `x=pi/3`