Extrema of a Function 1

EXTREMA OF A FUNCTION

Extrema of a function are points on the graph of the function where the function reaches its local maximums or minimums. In other words, these are points where the function is highest or lowest in its immediate vicinity.

HOW DO WE IDENTIFY EXTREMA?

Extrema of a function can be found using derivatives. If the derivative of a function f(x) is equal to zero at a point x = a, and if the sign of the derivative changes when passing through this point, then the point x = a is an extremum of the function.

PROCEDURE FOR FINDING EXTREMA:

1. FIND THE DERIVATIVE OF THE FUNCTION: First, calculate the first derivative of the function f(x), denoted as f'(x).
2. DETERMINE CRITICAL POINTS: Critical points are those where f'(x) = 0 or where f'(x) is not defined. (The article focuses on f'(x)=0, which are stationary points).
3. CHECK THE NATURE OF THE EXTREMUM: Using the second derivative test or the first derivative test, determine whether the critical point is a maximum, minimum, or an inflection/saddle point.

EXAMPLE:

Let's take the function f(x) = x³ – 3x² + 2.

1. FIND THE DERIVATIVE:
   f'(x) = 3x² – 6x.
2. DETERMINE CRITICAL POINTS: Solve f'(x) = 0.
   3x² – 6x = 0
   x(3x – 6) = 0
   The critical points are x = 0 and x = 2.
3. CHECK THE NATURE OF THE EXTREMUM (USING THE FIRST DERIVATIVE TEST):
   We examine the sign of f'(x) for values less than, between, and greater than the critical points:
4. • When x < 0 (e.g., x = -1): f'(-1) = 3(-1)² - 6(-1) = 3 + 6 = 9. Since f'(x) is positive, the function is increasing.
   • When 0 < x < 2 (e.g., x = 1): f'(1) = 3(1)² - 6(1) = 3 - 6 = -3. Since f'(x) is negative, the function is decreasing.
     Therefore, at x = 0, there is a local maximum.
   • When x > 2 (e.g., x = 3): f'(3) = 3(3)² - 6(3) = 27 - 18 = 9. Since f'(x) is positive, the function is increasing again.
     Therefore, at x = 2, there is a local minimum.