Chapter 4: Graphing and Optimization – Business Calculus Study Notes

Graphing and Optimization

First Derivative and Graphs

The first derivative of a function provides critical information about the function's increasing and decreasing behavior. By analyzing the sign of the derivative, we can determine where the function rises, falls, or remains constant.

• Increasing Intervals: A function f is increasing on an interval if for any two points x1 and x2 in the interval, with x1 < x2, we have f(x2) > f(x1).

• Decreasing Intervals: A function f is decreasing on an interval if f(x2) < f(x1) for x1 < x2.

• Constant Intervals: The function does not change value over the interval.

• Relationship to Derivative: If f'(x) > 0 on an interval, f is increasing there. If f'(x) < 0, f is decreasing.

Critical Numbers and Local Extrema

Critical numbers are values in the domain of f where f'(x) = 0 or f'(x) does not exist. These points are candidates for local maxima, minima, or points of inflection.

• Local Maximum: f has a local maximum at c if f(x) ≤ f(c) for all x near c.

• Local Minimum: f has a local minimum at c if f(x) ≥ f(c) for all x near c.

• First Derivative Test:

  • If f' changes from positive to negative at c, f has a local maximum at c.

  • If f' changes from negative to positive at c, f has a local minimum at c.

  • If f' does not change sign, there is no local extremum at c.

Second Derivatives and Concavity

The second derivative, f''(x), provides information about the concavity of a function and helps identify points of inflection.

• Concave Upward: If f''(x) > 0 on an interval, the graph of f is concave upward (shaped like a cup).

• Concave Downward: If f''(x) < 0 on an interval, the graph is concave downward (shaped like a cap).

• Inflection Point: A point where the concavity changes (from up to down or vice versa) and f is continuous.

f''(x)	f'(x)	Graph of y = f(x)	Examples
+	Increasing	Concave upward	Curves opening up
-	Decreasing	Concave downward	Curves opening down

Second Derivative Test for Local Extrema

The second derivative test helps classify critical points as local maxima or minima.

f'(c)	f''(c)	Graph of f is:	f(c)	Example
0	+	Concave upward	Local minimum	U-shaped curve
0	-	Concave downward	Local maximum	n-shaped curve
0	0	?	Test does not apply	Indeterminate

Summary of Key Procedures and Theorems

• Increasing/Decreasing Test: For interval (a, b):

  • If f'(x) > 0, f(x) is increasing.

  • If f'(x) < 0, f(x) is decreasing.

• Concavity Test: For interval (a, b):

  • If f''(x) > 0, f(x) is concave upward.

  • If f''(x) < 0, f(x) is concave downward.

• Extreme Value Theorem (EVT): A continuous function on a closed interval [a, b] attains both an absolute maximum and minimum.

• Procedure for Finding Absolute Extrema:

  1. Check continuity on [a, b].

  2. Find critical numbers in (a, b).

  3. Evaluate f at endpoints and critical numbers.

  4. Largest value: absolute maximum; smallest: absolute minimum.

• Second Derivative Test: If f'(c) = 0 and f''(c) > 0, f(c) is a local minimum; if f''(c) < 0, f(c) is a local maximum.

Additional info:

• Image 4 is a blank coordinate grid and is not directly relevant to the above explanations, so it is not included.