Math Problem Statement

На рисунке изображен график производной функции f(x), определенной на интервале (-3; 8). Найдите промежутки возрастания функции f(x). В ответе укажите сумму целых точек, входящих в эти промежутки.

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 f(xintervals)f(x intervals) is **.

increasing---

**### where its Step derivative ( f-by'(-stepx Explanation:

  1. .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 xx-axis).

  1. **Analysis ofx) the graph ) (:derivative):

  2. The function** f(x)>0f'(x) > 0 (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 )<x) < x-axis ).0 These intervals are : ) (negative) where the curve - lies ( (-3, ** -below2** the) \ () x \ - ()- (-axis1,.

1---

Step )2: Identify the intervals

From \ the graph:

  • f(x)>0f'(x) > 0 on the intervals:
    • ( (-3, -)
    • (13,5)(13, 5)
  1. **Integer points within) these )
    • (1intervals,3)(1 intervals, 3)
    • (5,7)(5, 7)

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---

Step14Step 14.

: - Sum In the integer points The integer ( points ( are3: ,\ [ 5-2, -1, 2,) ), the integer points are 3,46,3,4 6, 7 and] Adding them: \ ( [ -5 \2).

+4 (-.1 ) + 2 +Summ ing3 the + integer points6 +: 7 = 2+015-2 + 015


Final Answer:

The + sum1 of + the integer4 points + in the intervals where 5=f(x)5 = f(x) 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.

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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