Math Problem Statement
Solution
The questionThe is asking question asks us to identify the intervals to where the determine function the intervals f(x where) the function \( f \(x)) is is increasing increasing based on, based on the graph the of graph its derivative of its derivative f'( \( fx)'( \x).). Then Here, is we a are structured to solution:
determine the sum---
of### integer Key points Insight: includedThe in these function is **.
increasing---
**### where its Step derivative ( f-by'(-stepx Explanation:
-
.0 \ Behavior of ( f(x)) ( \above the): ( x - \ A)- functionaxis ().
f###(x Step) \1): Analyze the is increasing graph From the provided graph of f'( when \( f'(x) > 0 (the derivative is above the -axis).
-
**Analysis ofx) the graph ) (:derivative):
-
The function** (positive) From the where the graph curve of lies **above** the f \('(x x) \,)- weaxis can. observe2 where . f The function'(x \( f'() > 0x (above the -axis ).0 These intervals are : ) (negative) where the curve - lies ( (-3, ** -below2** the) \ () x \ - ()- (-axis1,.
1---
Step )2: Identify the intervals
From \ the graph:
- on the intervals:
- ( (-3, -)
- **Integer points within) these )
Step 3: Find: the integer points in these intervals**
- - (-3 In, - \(1) (-:3 The, integer - points are2 ,2, the -1 integer point \ is).
- ( ( - (21, \ ). -3) ): The integer points are ( In 2, 3 ( \ (-).
- (1,5 , 17)) \:), The integer the integer points points are are6 \(, 07 \).
and---
.
: - Sum In the integer points The integer ( points ( are3: ,\ [ 5-2, -1, 2,) ), the integer points are 7 and] Adding them: \ ( [ -5 \2).
+4 (-.1 ) + 2 +Summ ing3 the + integer points6 +: 7 =
Final Answer:
The + sum1 of + the integer4 points + in the intervals where is increasing is8:
\ ] Let me know if$$ --- ### Final Answer you: needThe further clarification or sum of the a integer detailed points breakdown is of any ** step! 8--- **###. Related Questions--- : Do1 you. What want a is detailed the analysis relationship of between the any sign part of? Let \( f me'( know! --- ### Related Questions: 1. What is the difference between $$f'(x) > 0 \x)$$ and the behavior of) $$f(x)$$? 2. How do you identify critical points from the graph of and $$\( f'( f'(xx)) <$$? 3. What happens to $$f(x)$$ when $$f'(x) = 0$$? 0 \4) for a. function $$f(x)$$? 2. How do critical points (where $$How f'(x do) = you0 determine intervals \ of)) decreasing influence for \( f(x \()$$' fs(x) behavior \? )? 35.. How can the How graph of can we determine \( conc favity'( usingx) \( \ f)'( helpx in) sketching \ \() f and(x \() \ f)? ''--- (###x Tip): Always \)? 4 remember. What that the ** arederivative $$the f implications'(x of)$$** $$f indicates the'( slopex of \( f(x)) \$$). being Positive $$constant f on'(x an) interval$$? means5 $$f. How(x) do$$ is we calculate areas increasing, of and regions negative $$enclosed f'( byx \() f \'() meansx \() f \(x)) \ and) is the \( decreasing. x$$-axis? --- ### Tip: Always pay attention to where $$f'(x)$$ crosses the $$x$$-axis—these are critical points where the behavior of $$f(x)$$ may change.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Derivative Analysis
Intervals of Increase
Formulas
f(x) is increasing when f'(x) > 0
Intervals of increase are determined by the sign of f'(x)
Theorems
First Derivative Test
Suitable Grade Level
Grades 10-12