91Ó°ÊÓ

Critical points are pivotal in calculus when analyzing functions for differentiability. They occur where the derivative of a function is zero or undefined. In our case, we examine the derivative of the function inside the absolute value expression, which is given by:

• \( f(x) = |x^3 - 9x| \)
• The derivative, ignoring the absolute value, is \( f'(x) = 3x^2 - 9 \).

To find critical points, we need to determine where \( f'(x) = 0 \) or is undefined. Solving \( 3x^2 - 9 = 0 \), we arrive at:

• \( x = \pm \sqrt{3} \)

However, the presence of the absolute value complicates matters, as the function inside it can cause non-differentiability at its roots. So, in addition to \( x = \pm \sqrt{3} \), we also consider points where \( x^3 - 9x = 0 \):

• \( x = 0, 3, -3 \)

These roots are critical because they represent sharp edges or corners in the absolute value function, where differentiability is lost.

Extrema refer to the maximum and minimum points of a function. For the absolute value function \( f(x) = |x^3 - 9x| \), we find extrema by evaluating the critical points found earlier. Extrema can be local, referring to peaks or troughs within a particular interval, or global, covering the entire domain of the function.To find these, we calculate \( f(x) \) at critical points:

• At \( x = 0 \), \( f(0) = |0^3 - 9 \times 0| = 0 \)
• At \( x = 3 \) or \( x = -3 \), \( f(x) = |x^3 - 9x| = 0 \)
• Evaluate \( f(\pm \sqrt{3}) \). The value of the derivative here indicates changes in the direction of the function, i.e., an extremum.

These evaluations confirm whether each point is a local minimum or maximum. Specifically:- At \( x = \pm \sqrt{3} \), cross-checking with the second derivative or a sign test on intervals can confirm their nature.This completes the identification of extrema for \( f(x) \), determining where the function achieves its highest or lowest values on the intervals considered.

The absolute value function is unique because it combines two pieces of linear functions, each affecting the continuity and differentiability of the function at certain points. With \( f(x) = |x^3 - 9x| \), we have:

• When \( x^3 - 9x \) is positive, the absolute value does not change the derivative.
• When \( x^3 - 9x \) switches to negative, the function's nature shifts, often leading to non-differentiable points.

This behavior is crucial at the roots, \( x = 0, 3, -3 \), where the inside expression equals zero. At these points, the function transitions from positive to negative or vice versa, creating a 'corner.' Such corners lead to discontinuity in the derivative, causing non-existence of \( f'(x) \) at these very points.To summarize:

• Absolute value causes changes in direction, termed as 'cusps.'
• This results in places where derivatives do not exist.

Recognizing these characteristics of absolute value functions helps in analyzing their graphs and understanding differentiability concerns.