Monotony

Monotony involves understanding whether a function is increasing, decreasing, or constant across specific intervals.

Determine Monotony

• If $f^{\prime }(x)>0$ for a given interval, the function is increasing within that interval.
• If $f^{\prime }(x)<0$ for a given interval, the function is decreasing within that interval.
• If $f^{\prime }(x)=0$ or undefined, it indicates potential extrema or points of inflection but doesn’t impact monotony directly.

Example

Let's consider the function f(x) = x^3 - 3x.

• The first derivative is $y^{\prime }=3x^2-3$

Next, we’ll set the derivative equal to zero to find the Critical Points: $3x^2-3=0$

Solving for $x$: $x^2-1=0$ $x^2=1$ $x=\pm 1$

• $f(x)$ is strictly increasing on $x<-1$ and $x>1$
• $f(x)$ is strictly decreasing on $x>-1$ and $x<1$