Question

Is `f(x)=cos(-x)` increasing or decreasing at `x=0`?

Answer 1

`f(x)=cos(-x)` is neither increasing nor decreasing at `x=0`

Explanation:

The first derivative provides the slope at each point of our function. A positive slope means the function is increasing and that a negative slope means that the function is decreasing.

Knowing this, we can find out if `cos(-x)` is increasing or decreasing at `color(red)(x=0)` by evaluating its first derivative at that point.

Before starting, we may want to simplify our function. Because `cos(x)` is an even function, we know that `cos(-x)=cos(x)`.

`d/dx[cos(x)]`

`= -sin(x)`

Evaluate at `color(red)(x=0)`:

`-sin(color(red)0)`

`= -0`

`= 0`

In this case, the slope is neither negative or positive, but 0. Hence, `f(x)=cos(-x)` is neither increasing nor decreasing at `color(red)(x=0)`.