Business Calculus

1. Business Calculus Extrema

2. Extrema: Basic Facts Two facts about the graph of a function will help us in seeing where extrema may occur. The intervals where the graph is rising or falling. The possible shapes of the graph at its highest or lowest points.

3. 2.1 Intervals of Increase and Decrease: • A function is increasing if the function is rising from • left to right. • A function is decreasing if the function is falling from • left to right. • A critical value of f is a point of the graph where fmight • change from increasing to decreasing or decreasing to • increasing. • Critical values are found in two ways: • f has a critical value at x = cif either of the following are true: • 1. • 2.

4. Intervals of increase and intervals of decrease: We can use derivatives to determine where a function is increasing and where it is decreasing. If f ′(x) > 0 for all x in an interval, then f is increasing on that interval. 2. If f ′(x) < 0 for all x in an interval, then f is decreasing on that interval.

5. 2.2 Concavity: • A function is concave up if the function is in the shape of an • upright bowl. • A function is concave down if the function is in the shape of an • upside down bowl. On a given interval, the second derivative will tell us whether the function is concave up or concave down. If f ʺ(x) > 0 for all x in an interval, then f is concave up on that interval. 2. If f ʺ(x) < 0 for all x in an interval, then f is concave down on that interval.

6. 2.1 Relative extrema: • A function f has a relative maximum at x = c if • 1. c is in the domain of f and • 2. f (c) is greater than all other y values of the function in an • interval containing c. • A function f has a relative minimum at x = c if • 1. c is in the domain of f and • 2. f (c) is less than all other y values of the function in an • interval containing c.

7. 2.4 Absolute extrema: • A function f has an absolute maximum at x = c if • 1. c is in the domain of f and • 2. f (c) is greater than all other y values of the function • on its entire domain. • A function f has an absolute minimum at x = c if • 1. c is in the domain of f and • 2. f (c) is less than all other y values of the function • on its entire domain.

8. Important Fact: relative and absolute extrema can occur only at critical values or endpoints of the function. English translation: “What is the relative (or absolute) extrema” means give a y value. “Where does the relative (or absolute) extrema occur” means give an x value. • To find relative and absolute extrema of a function: Find all critical points and endpoints. This is the list of possible extrema. B) Test each critical point and endpoint to determine if it is a minimum, maximum, or neither one.

9. Testing Critical Values for Relative Extrema There are two ways to test critical points for relative extrema. 2.1 First Derivative Test: Note: A function has a relative minimum at a point in its domain if it changes from decreasing to increasing at that point. A function has a relative maximum at a point in its domain if it changes from increasing to decreasing at that point. Because the derivative can tell us about the direction of the curve, we can use the derivative to test critical values to determine if it gives a relative minimum, relative maximum, or neither one.

10. First Derivative Test • f has a relative minimum at the critical point x = c if • 1. c is in the domain of f and • 2. f ′(x) < 0 for x on the left of c and f ′(x) > 0 for x on the right of c. • f has a relative maximum at the critical point x = c if • 1. c is in the domain of f and • 2. f ′(x) > 0 for x on the left of c and f ′(x) < 0 for x on the right of c. • We will use a number line to organize this information and • find relative extrema at critical points and endpoints.

11. 2.2 Second Derivative Test: Note: If a critical value comes from setting y′ = 0, then it sits at either the top of an upside down bowl (concave down), or the bottom of an upright bowl (concave up). We can use this information to determine if a critical value is a relative minimum or relative maximum, but only if the critical value comes from setting y′ = 0.

12. Second Derivative Test • f has a relative minimum at the critical point x = c if • 1. c is in the domain of f found by f ′(x) = 0 and • 2. f ʺ(c) > 0 • f has a relative maximum at the critical point x = c if • 1. c is in the domain of f found by f ′(x) = 0 and • 2. f ʺ(c) < 0 • It is important to note that we are plugging c into the second • derivative to test whether x = c will give a minimum or maximum.

13. Testing Critical Values for Absolute Extrema There are two ways to test critical values and endpoints for absolute extrema, the extreme value theorem and a special case of the second derivative test. • Extreme Value Theorem (Principal 1) • If f is continuous on a closed interval [a, b], then f will attain • its absolute minimum and its absolute maximum. • This means that if we make a list of all critical values and • endpoints (x values), and then find each corresponding y value, • the highest y value is the absolute maximum and the lowest y • value is the absolute minimum.

14. Using the 2nd derivative test for absolute extrema • If we are interested in a continuous function, and it has only one • critical value which is found by solving y′ = 0, then we can use • the second derivative test to determine absolute extrema at that • critical value. • Note: it is not required that we are working on a closed interval. Since there is only one critical value, this value will give either an absolute minimum or an absolute maximum. For a single critical value of x = c which is in the domain of f and comes from y′ = 0 1. f will have an absolute minimum at x = c if f ʺ(c) > 0. 2. f will have an absolute minimum at x = c if f ʺ(c) > 0. • 2nd derivative test (Principal 2)