This document discusses key concepts related to finding extrema of functions, including: - Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a subinterval, respectively. - Critical points, where the derivative is zero or undefined, and endpoints must be checked to find extrema. - The Extreme Value Theorem states that continuous functions on closed intervals have both a maximum and minimum value. - The first derivative determines whether a function is increasing or decreasing, and where it is zero may indicate relative extrema. The second derivative indicates concavity and points of inflection.

1. By Nofal Umair

2. Extreme values are termed “extrema”
Absolute Extrema: the point in question represents either the
maximum or minimum value of the function over the domain.
Relative Extrema: the point in question represents either the
maximum or minimum value of the function on a specified
segment of the domain (or in the “hood”).
Let’s take a look at an example and consider several points on a
function.

3. Take a look at this diagram. Notice the difference between
local and absolute extrema.

4. Although the concepts are easy to understand, be attuned to
the subtlety of the questions asked …
Example 1: on the closed interval find any relative extrema for
y = sin x and y = cos x
Example 2: Same as above but on the open interval
,
2 2
  
  
,
2 2
  
 
 

5. The Extreme Value Theorem:
If a function f is continuous on a closed interval [a, b], then f has both a
maximum and a minimum on the interval.
Question: So, by looking at a graph of a function, how can you find its
extrema?
What’s true about the
curve at its max and
min?
What’s the one
analytical tool we’ve
been studying all year?

6. To fully answer the question, we need to define some terms
and give you one theorem.
Critical Points – a point on the interior of the domain of a function f at
which
○ f’ = 0 or
○ f’ does not exist (is undefined).
Theorem: if a function f has a local maximum and/or minimum at
some interior point “c” of its domain, and if f’(c) exists, then
f’(c) = 0
SO, HOW DO YOU FIND EXTREMA???
1. Find all critical points (values)
2. Check the endpoints of the specified domain

7. A little Practice
Find the extrema of on the interval [-1, 2].
1. Find the critical numbers in f
2. Evaluate f at each critical number
3. Evaluate f at each endpoint
4. Compare. Least number is minimum, greatest number is maximum.
Answers are:
1. Critical numbers at x = 0 and x = 1
2. f(0) = 0 and f(1) = -1
3. Left endpoint has height of 7, right endpoint has height of 16
4. Min = -1, max = 16
4 3
( ) 3 4f x x x 

8. Why Need Of Derivative??????
 Derivatives have arisen from the need to manage the
risk arising from movements in markets beyond our
control ,which may severely impact the revenues and
costs of the firm.
 Derivatives can be used in a number of ways in
everyday life, especially with optimization.
Example:
The growth rate of any company , the profit or loss
it made, etc

9. CURVES
 You can easily graph any function by
knowing three things.
 1) ZEROS AND UNDEFINED SPOTS
 2) MAXIMUM AND MINIMUM
POINTS
 3) CONCAVITYAND INFLECTION
POINTS.

10. What the First Derivative Tells Us:
 Suppose that a function f has a derivative at
every point x of an interval I. Then:
increases on I if ( ) 0 for all in I.f f x x 
decreases on I if ( ) 0 for all in I.f f x x 

11. What This Means:
 In geometric terms, the first derivative tells
us that differentiable functions increase on
intervals where their graphs have positive
slopes and decrease on intervals where their
graphs have negative slopes.
 WHAT HAPPENS IF THE FIRST
DERIVATIVE IS ZERO?

12. When The First Derivative is
Zero
 A derivative has the intermediate value
property on any interval on which it is
defined.
 It will take on the value zero when it
changes signs over that interval.
 Thus, when the derivative changes signs on
an interval, the graph of f(x) must have a
horizontal tangent.

13. Relative Maxima and Minima
 If the derivative
changes sign, there
may be a local max or
min, as shown here.
 More on this later.

14. Concavity
 Concave down—”spills water”
 Concave up—”holds water”
 The graph of
is concave down on any interval where
and concave up on any interval where
( )y f x
0y 
0y 

15. Points of Inflection
 A point on the curve where the concavity changes
is called a point of inflection.
 If the second derivative is zero for some x, we
may be able to find a point of inflection.
 It IS possible for the second derivative to be zero
at a point that is NOT a point of inflection.
 A point of inflection may occur where the second
derivative fails to exist.

16. Inflection Points
 You can tell where the function changes concavity
by finding the inflection points.
 Evaluate the function at those values where the
second derivative is zero;
 Take a look at the graph of the original function:
4 2( ) 2f x x x 

17. The Graph

18. An Interesting Example
Suppose that the yield, r, in the % of students in a one
hour exam is given by:
r = 300t (1−t).
Where 0 < t < 1 is the time in hours.
1. At what moments is the yield zero?
2. At what moments does the yield increase or
decrease?
3. When is the biggest yield obtained and which is?

19. ERRORS AND APPROXIMATIONS
 We can use differentials to calculate small changes in
the dependent variable of a function corresponding
to small changes in the independent variable.
e.g