Math  /  Calculus

Question(7 pts) Let f(x)f(x) be a twice differentiable function and the graph of the derivative f(x)f^{\prime}(x) is given below. i) f(x)f(x) is increasing on the interval (1,2)(1,2). ii) f(x)f(x) is decreasing on the interval (2,5)(2,5). iii) (5,f(5))(5, f(5)) is a critical point of f(x)f(x). iv) (3.5,f(3.5))(3.5, f(3.5)) is an inflection point of f(x)f(x).
Choose all the correct statements

Studdy Solution

STEP 1

1. f(x) f(x) is a twice differentiable function.
2. The graph of the derivative f(x) f'(x) is given.
3. We need to determine which statements about f(x) f(x) are correct based on the graph of f(x) f'(x) .

STEP 2

1. Analyze the intervals where f(x) f(x) is increasing or decreasing.
2. Determine the critical points of f(x) f(x) .
3. Identify the inflection points of f(x) f(x) .

STEP 3

To determine where f(x) f(x) is increasing, look for intervals where f(x)>0 f'(x) > 0 .
The graph shows f(x)>0 f'(x) > 0 between x=1 x = 1 and x=2 x = 2 , so f(x) f(x) is increasing on (1,2) (1, 2) .

STEP 4

To determine where f(x) f(x) is decreasing, look for intervals where f(x)<0 f'(x) < 0 .
The graph shows f(x)<0 f'(x) < 0 between x=2 x = 2 and x=5 x = 5 , so f(x) f(x) is decreasing on (2,5) (2, 5) .

STEP 5

A critical point occurs where f(x)=0 f'(x) = 0 or f(x) f'(x) is undefined.
The graph shows f(x)=0 f'(x) = 0 at x=5 x = 5 , so (5,f(5)) (5, f(5)) is a critical point.

STEP 6

An inflection point occurs where the concavity changes, which is indicated by a change in the sign of f(x) f''(x) or where f(x) f'(x) changes from increasing to decreasing or vice versa.
The graph shows a change in the slope of f(x) f'(x) at x=3.5 x = 3.5 , indicating an inflection point at (3.5,f(3.5)) (3.5, f(3.5)) .
The correct statements are: i) f(x) f(x) is increasing on the interval (1,2) (1,2) . ii) f(x) f(x) is decreasing on the interval (2,5) (2,5) . iii) (5,f(5)) (5, f(5)) is a critical point of f(x) f(x) . iv) (3.5,f(3.5)) (3.5, f(3.5)) is an inflection point of f(x) f(x) .
All statements are correct.

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