MAT 213Brief Calculus

Section 4.2Relative and Absolute Extreme Points

• Let’s see what we remember about derivatives of a function and its graph– If f’ > 0 on an interval than f is

• Increasing

– If f’ < 0 on an interval than f is• Decreasing

– If f’(a) = 0 than f is• Neither increasing nor decreasing

• Has a horizontal tangent at a

Use the derivative to determine where each function is increasing and where it is

decreasing. Verify by looking at the graphs.

f(x)=x2 – 4x + 7

g(x)=x3 - 3x2

In the previous example, f(x)=x2 - 4x + 7,

the point x=2 is called a critical point because f’(2) = 0

Critical Points

• For any function f, a point p in the domain of f where f’(p) = 0 or f’(p) is undefined is called a critical point of the function – The critical value of f is the function value, f(p)

where p is the critical point– Critical points are used to determine relative

extrema

Relative Extrema

• f has a local (relative) maximum at x = p if f(p) is equal to or larger than all other f values near p– If p is a critical point and f’ changes from positive to

negative at p, then f has a local maximum at p

• f has a local (relative) minimum at x = p if f(p) is equal to or smaller than all other f values near p– If p is a critical point and f’ changes from negative to

positive at p, then f has a local maximum at p

• Since in the previous cases we were using the first derivative, we were using the first derivative test to check for relative extrema

The First Derivative Test for Local Maxima and Minima

Suppose p is a critical point of a continuous function f.

If f’ changes from negative to positive at p, then f has a local minimum at p.

If f’ changes from positive to negative at p, then f has a local maximum at p.

• Often we want to find the maximum or minimum value of a function

• This is called optimization

• Absolute (Global) Maxima and Minima– f has an absolute (or global) minimum at p if f(p)

is less than or equal to all values of f– f has a global maximum at p if f(p) is greater than

or equal to all values of f

• Absolute (or global) maxima and minima are sometimes referred to as “extrema” or “optimal values”

• We may not always have a relative extrema and we may not always have absolute extrema

• Let’s take a look at the following graphs and identify– Relative and absolute maxima– Relative and absolute minima– Concavity– Inflection points

• The number of Nonbusiness Chapter 11 Bankruptcies can be modeled by a cubic polynomial

• The cubic would have the following equation with the plot on the next slide

Years (Since1998)

Bankruptcies

0 981

1 731

2 722

3 745

4 894

5 966

6 935

Consider the following situation

B(t) = −12.58t3+ 134.1t2− 358.8t + 979.2

• Where are the relative extrema in this graph?

• Where are the absolute extrema in this graph?

• Now in the previous graph we had relative extrema, but no absolute extrema

• Recall that our model was from 1998 (or t = 0) to 2004 (or t = 6)

• Thus we must have absolute extrema between those years

• Let’s take a look at the graph from that point of view and see if we can identify any absolute extrema

Extreme Value Theorem• If f is a continuous function on the closed

interval a ≤ x ≤ b, then f has a global maximum and a global minimum on that interval

• If f is continuous over a closed interval, we are guaranteed to have an absolute max and absolute min

• They either occur at critical points or at the endpoints

• Thus the procedure is to find all critical points of f, evaluate f at the critical points and endpoints and compare values

• To find the absolute extrema of a continuous function on a closed interval (i.e. endpoints are included):– Compare function values at the critical points and

endpoints

• To find the absolute of a continuous function on an open interval (i.e. endpoints not included or infinite endpoint):– Find the value of the function at all critical points and

sketch the graph.– Look at the function values when x approaches the

endpoints of the interval, or approaches ±∞, when appropriate.

• Let’s try 27 from the book