Extreme points are points where a function attains its maximum or minimum value. To find the extreme values of a function f(x), take the derivative f'(x) and set it equal to 0, then solve for x to find the x-coordinates of local maxima and minima. The y-coordinates are found by evaluating the original function f(x) at those x-values. A minimum value is the lowest y-value of a function, while a maximum is the highest y-value. Functions can have multiple or infinitely many maxima and minima.

1. Extreme point
By :
Rozhin Mehdi
Hevi serwan
Supervised by :
Dr.Rafiq Saleh
6/1/2021 extreme point 1
UNIVERSITY OF GARMIAN
CIVIL ENGINEERING
DEPARTMENT
FIRST STAGE

2. 1. What are extreme points math?
2. How do you find the extreme values
of the function and where they
occur?
3. Sample Problem.
4. Sources.
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Content
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3. What are extreme points
math?
• An extreme point, in mathematics, is a
point in a convex set which does not lie in
any open line segment joining two points in
the set. Extreme point or extremal point
may also refer to: A point where some
function attains its extremum. A leaf vertex
of a tree in graph theory.
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4. What are extreme
points math?
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5. How do you find the extreme values of
the function and where they occur?
Explanation:
To find extreme values of a function
f , set f ' ( x ) = 0 and solve. This
gives you the x-coordinates of the
extreme values/ local maxs and
mins.
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6. For example:
consider f ( x ) = x 2 − 6 x + 5 . To find the minimum
value of f (we know it's minimum because the
parabola opens upward), we set f ' ( x ) = 2 x − 6 = 0
Solving, we get x = 3 is the location of the minimum.
To find the y-coordinate, we find f ( 3 ) = − 4 .
Therefore, the extreme minimum of f occurs at the
point ( 3 , − 4 ) .
How do you find the extreme values of
the function and where they occur?
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7. Extreme points
Extreme points, also called extrema,
are places where a function takes on an
extreme value that is, a value that is
especially small or especially large in
comparison to other nearby values of
the function. Extrema look like the tops
of hills and the bottoms of valleys.
Time to go hiking
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8. There are two types of extreme
points:
1. minima (the valleys).
2. maxima (the hills).
Extreme points
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9. • Extreme points can be local or global.
• We need to define minimum and maximum
values without the on an interval bit:
1. A minimum value of a function is a y-
value of the function that is as low, or
lower, than other values of the function
nearby. A minimum looks like a valley:
Extreme points
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10. Sample Problem
The minimum value of the
function f (x) = x2 + 1 is
y = 1:
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11. The minimum value of the
function f (x) = cos x is y = -1:
Sample Problem
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12. 2. A function may have multiple
maxima.
Extreme points
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13. The function graphed below has
two maxima: y = 2 and y = 3.
Sample Problem
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14. • A function may have infinitely many
maxima.
• The function graphed below has
infinitely many maxima:
Extreme points
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16. Sources
1. https://socratic.org/
2. https://www.shmoop.com/
3. www.Wikipedia.com
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