0 A Review Of Basic Algebra 1 Equations And Inequalities 2 Functions And Graphs 3 Functions 4 Polynomial And Rational Functions 5 Exponential And Logarithmic Functions 6 Linear Systems 7 Conic Sections And Quadratic Systems 8 Sequences, Series, And Probability Chapter3: Functions
3.1 Graphs Of Functions 3.2 Transformations Of The Graphs Of Functions 3.3 More On Functions; Piecewise-defined Functions 3.4 Operations On Functions 3.5 Inverse Functions 3.CR Chapter Review 3.CT Chapter Test 3.CM Cumulative Review Exercises Section3.3: More On Functions; Piecewise-defined Functions
Problem 1SC: Classify the function as being even, odd, or neither. Problem 2SC Problem 3SC Problem 4SC Problem 5SC Problem 6SC Problem 7SC Problem 8SC Problem 9SC Problem 10SC Problem 1E: Fill in the blanks. If the graph of a function is symmetric about the ___________, it is called an... Problem 2E Problem 3E Problem 4E Problem 5E: Fill in the blanks. If the values of fx get larger as x increases on an interval, we say that the... Problem 6E: Fill in the blanks. If the values of fx get smaller as x increases on an interval, we say that the... Problem 7E: Fill in the blanks. If the values of fx do not change as x increases on an interval, we say that the... Problem 8E Problem 9E: Fill in the blanks. A local ________ occurs where a function changes from decreasing to increasing. Problem 10E Problem 11E: Determine whether each function even, odd, or neither. Problem 12E Problem 13E: Determine whether each function even, odd, or neither. Problem 14E: Determine whether each function even, odd, or neither. Problem 15E: Determine whether each function even, odd, or neither. Problem 16E: Determine whether each function even, odd, or neither. Problem 17E: Determine algebraically whether each function is even, odd, or neither. fx=x4+x2 Problem 18E: Determine algebraically whether each function is even, odd, or neither. fx=x3-2x Problem 19E: Determine algebraically whether each function is even, odd, or neither. fx=x3+x2 Problem 20E: Determine algebraically whether each function is even, odd, or neither. fx=x6-x2 Problem 21E: Determine algebraically whether each function is even, odd, or neither. fx=x5+x3 Problem 22E: Determine algebraically whether each function is even, odd, or neither. fx=x3-x2 Problem 23E: Determine algebraically whether each function is even, odd, or neither. fx=2x3-3x Problem 24E: Determine algebraically whether each function is even, odd, or neither. fx=4x2-5 Problem 25E: Determine algebraically whether each function is even, odd, or neither. fx=xx2-1 Problem 26E: Determine algebraically whether each function is even, odd, or neither. fx=2xx2-9 Problem 27E: Determine algebraically whether each function is even, odd, or neither. fx=1x4 Problem 28E: Determine algebraically whether each function is even, odd, or neither. fx=-2x2 Problem 29E: Determine algebraically whether each function is even, odd, or neither. fx=x+1 Problem 30E: Determine algebraically whether each function is even, odd, or neither. fx=2x-5 Problem 31E: Determine algebraically whether each function is even, odd, or neither. fx=xx Problem 32E: Determine algebraically whether each function is even, odd, or neither. fx=2x-x Problem 33E: State the open intervals where each function is increasing, decreasing, or constant. Problem 34E: State the open intervals where each function is increasing, decreasing, or constant. Problem 35E: State the open intervals where each function is increasing, decreasing, or constant. Problem 36E: State the open intervals where each function is increasing, decreasing, or constant. Problem 37E: State the open intervals where each function is increasing, decreasing, or constant. Problem 38E: State the open intervals where each function is increasing, decreasing, or constant. Problem 39E: Use the graph to identify any local maxima and local minima. Problem 40E: Use the graph to identify any local maxima and local minima. Problem 41E: Use the graph to identify any local maxima and local minima. Problem 42E: Use the graph to identify any local maxima and local minima. Problem 43E: Use the graph to identify any local maxima and local minima. Problem 44E: Use the graph to identify any local maxima and local minima. Problem 45E: Use the graph to identify any local maxima and local minima. Problem 46E: Use the graph to identify any local maxima and local minima. Problem 47E: Use the graph to identify any local maxima and local minima. Problem 48E: Use the graph to identify any local maxima and local minima. Problem 49E: Evaluate each piecewise-defined function. fx=2x+2ifx03ifx0 a. f-2 b. f0 Problem 50E: Evaluate each piecewise-defined function. fx=x-2ifx1x2ifx1 a. f-1 b. f5 Problem 51E: Evaluate each piecewise-defined function. fx=2ifx02-xif0x2x+1ifx2 a. f-1 b. f1 c. f2 Problem 52E: Evaluate each piecewise-defined function. fx=2xifx03-xif0x2xifx2 a. f-0.5 b. f0 c. f2 Problem 53E: Graph each piecewise-defined function. fx=x+2ifx<02ifx0 Problem 54E: Graph each piecewise-defined function. fx=2xifx<0-2xifx0 Problem 55E: Graph each piecewise-defined function. fx=xifx02ifx0 Problem 56E: Graph each piecewise-defined function. fx=-xifx<012xifx>0 Problem 57E: Graph each piecewise-defined function. fx=-4-xifx13ifx1 Problem 58E: Graph each piecewise-defined function. fx=-5-xifx1-3ifx1 Problem 59E: Graph each piecewise-defined function. fx=-xifx<0x2ifx0 Problem 60E Problem 61E Problem 62E Problem 63E Problem 64E Problem 65E Problem 66E Problem 67E Problem 68E Problem 69E Problem 70E: Graph each function. y=x+2 Problem 71E Problem 72E Problem 73E Problem 74E: Riding in a taxi A taxicab company charges 3 for a trip upto 1 mile, and 2 for every extra mile or... Problem 75E Problem 76E: iPad repair There is a charge of 30, plus 40 per hour or fraction of an hour, to repair an iPad.... Problem 77E: Rounding numbers Measurements are rarely exact; they are often rounded to an appropriate precision.... Problem 78E Problem 79E Problem 80E Problem 81E Problem 82E Problem 83E Problem 84E Problem 85E: Describe what happens at the point where the graph of a function changes from increasing to... Problem 86E Problem 87E Problem 88E Problem 89E Problem 90E Problem 91E Problem 92E Problem 93E Problem 94E Problem 95E Problem 96E: Determine if the statement is true or false. If the statement is false, then correct it and make it... Problem 97E Problem 98E: Determine if the statemment is true or false. If the statement is false, then correct it and make it... Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it... Problem 100E: Determine if the statement is true or false. If the statement is false, then correct it and make it... Problem 97E
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PROOF THE STATEMENT FOR LOCAL MINIMUM POINT IS ALSO TRUE PLEASE ???
Transcribed Image Text: Theorem :
Let f be defined on [a, b] , if f has a local maximum at a
point x E (a, b), and if f'(x) exists, then f'(x) = 0.
The statement for local minimum point is also true.
Proof :
Let 8 > 0,
a <x - 8 < x < x+ 8<b
If
x - 8<t<x
then
f(t) < f(x),t– x < 0
f(t) - f(x)
t-x
f(t) - f(x)
lim
t-x
: f'(x) 2 0
(1)
If
x <t< x+8 then
f(t) < f(x),
t- x>0
f(t) - f(x)
<0
t-x
f(t)-f(x)
lim
t-x
: f'(x) <0
: f'(x) < 0..
(2)
Hence from (1) and (2 ) we conclude that f'(x) = 0
Formula Formula A function f ( x ) is also said to have attained a local minimum at x = a , if there exists a neighborhood ( a − δ , a + δ ) of a such that, f ( x ) > f ( a ) , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a f ( x ) − f ( a ) > 0 , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a In such a case f ( a ) is called the local minimum value of f ( x ) at x = a .
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![Theorem :
Let f be defined on [a, b] , if f has a local maximum at a
point x E (a, b), and if f'(x) exists, then f'(x) = 0.
The statement for local minimum point is also true.
Proof :
Let 8 > 0,
a <x - 8 < x < x+ 8<b
If
x - 8<t<x
then
f(t) < f(x),t– x < 0
f(t) - f(x)
t-x
f(t) - f(x)
lim
t-x
: f'(x) 2 0
(1)
If
x <t< x+8 then
f(t) < f(x),
t- x>0
f(t) - f(x)
<0
t-x
f(t)-f(x)
lim
t-x
: f'(x) <0
: f'(x) < 0..
(2)
Hence from (1) and (2 ) we conclude that f'(x) = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e35a088-ac88-4bc6-8102-24be224e730b%2Fc872a70c-d14b-49b6-8a6f-5dcf4a40303d%2F5dlesaw_processed.jpeg&w=3840&q=75)
