Find critical and stationary points f if f(x)=|sin⁡x |.also find relative extrema

To find the critical and stationary points of the function f(x) = |sin(x)|, we first need to understand that the absolute value function |sin(x)| is not differentiable at x = nπ where n is an integer. This is because the graph of |sin(x)| has sharp turns at these points.

1. Critical Points: Critical points of a function are where its derivative is zero or undefined. The derivative of |sin(x)| is undefined at x = nπ (n is an integer) because of the sharp turns in the graph at these points. So, the critical points of f(x) = |sin(x)| are x = nπ.

2. Stationary Points: Stationary points of a function are where its derivative is zero. However, the derivative of |sin(x)| is never zero because |sin(x)| always increases or decreases. Therefore, f(x) = |sin(x)| has no stationary points.

3. Relative Extrema: Relative extrema of a function are the local minimum and maximum points. For f(x) = |sin(x)|, the local minimum occurs at x = nπ (n is an integer) where the value of the function is 0. The local maximum occurs at x = π/2 + nπ (n is an integer) and x = -π/2 + nπ (n is an integer) where the value of the function is 1. So, the relative extrema of f(x) = |sin(x)| are at x = nπ (minimum) and x = π/2 + nπ, -π/2 + nπ (maximum).