A.P. Calculus 3.1: Extrema on an Interval

A.P. Calculus 3.1: Extrema on an Interval

Definition of Extrema

• A function can have both maximum and minimum values on a closed interval.

  • Example: For the function $f(x) = x^2 + 1$, it has both a minimum and a maximum on the closed interval $[-1, 2]$.

• The function does not have a maximum on the open interval $(-1, 2)$.

Extreme Value Theorem

• This theorem states that if a function is continuous on a closed interval, it must attain a maximum and minimum value on that interval.

Relative Extrema

• Relative maxima and minima can be understood in terms of a graph.

  • A relative maximum occurs at a point that resembles the top of a hill.

  • A relative minimum occurs at a point that resembles the bottom of a valley.

• To determine relative extremum, one can find the value of the derivative at each point identified as a relative extremum (as illustrated in Figure 3.3).

Critical Numbers

• Critical Numbers: At each relative extremum, the x-values where the derivative is either zero or does not exist are referred to as critical numbers.

• Critical numbers are important because they indicate possible locations of relative extrema.

Finding Extrema on a Closed Interval

• You Try: To practice, find the extrema of the function $f(x) = 3x^4 – 4x^3$ on the closed interval $[-1, 2]$.

Additional Notes on Critical Numbers

• In the example shown in Figure 3.5, it is noted that the critical number $x = 0$ does not yield a relative minimum or maximum.

• This indicates that the converse of Theorem 3.2 is not necessarily true: critical numbers of a function do not always produce relative extrema.

Communication of Solutions

• It is essential to document the communication of your solutions, thoughts, and thought process clearly.

• When solving problems, label the page/source and each problem with its specific number.

• Assigned problems from the textbook (p. 171): 9, 12, 14, 15, 19, 21, 26, 29, 31, 39, 43, 47, 48, 49, 57, 62, 64.