Increasing and Decreasing Functions

A function is strictly increasing on an interval if \(x_1<x_2\Rightarrow f(x_1)<f(x_2)\), and strictly decreasing if \(x_1<x_2\Rightarrow f(x_1)>f(x_2)\). For differentiable functions:

• \(f\,'(x)>0\) for all \(x\) in an open interval ⇒ \(f\) is strictly increasing there;
• \(f\,'(x)<0\) ⇒ strictly decreasing;
• \(f\,'(x)=0\) throughout ⇒ constant.

Finding the intervals

1. Find \(f\,'(x)\) and solve \(f\,'(x)=0\) (these points split the real line).
2. Test the sign of \(f\,'(x)\) in each interval.
3. Positive ⇒ increasing, negative ⇒ decreasing.

Example: \(f(x)=x^3-6x^2+9x+1\), \(f\,'(x)=3(x-1)(x-3)\). Increasing on \((-\infty,1]\) and \([3,\infty)\), decreasing on \([1,3]\).