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Math  /  Calculus

Question22 W Mark for Review ape
The function ff is differentiable and increasing on the interval 0≤x≤60 \leq x \leq 6, and the graph of ff has exactly two points of inflection on this interval. Which of the following could be the graph of f′f^{\prime}, the derivative of ff ? (A) (B) (C) (D)

Studdy Solution

STEP 1

1. The function f f is differentiable and increasing on the interval 0≤x≤6 0 \leq x \leq 6 .
2. The graph of f f has exactly two points of inflection on this interval.
3. We need to determine which graph could represent f′ f' , the derivative of f f .

STEP 2

1. Understand the implications of f f being increasing and having points of inflection.
2. Analyze the behavior of f′ f' based on the characteristics of f f .
3. Match the behavior of f′ f' with the given graph options.

STEP 3

Since f f is increasing on 0≤x≤6 0 \leq x \leq 6 , f′(x) f'(x) must be non-negative on this interval.
Points of inflection occur where the concavity of f f changes, which means f′′(x) f''(x) changes sign.

STEP 4

For f f to have two points of inflection, f′ f' must change from increasing to decreasing and then back to increasing (or vice versa) twice.
This means f′ f' should have two turning points, indicating changes in concavity.

STEP 5

Graph A: Steadily increasing, no turning points. Graph B: Increases to a peak, then decreases, one turning point. Graph C: Increases, decreases, then increases again, two turning points. Graph D: Two peaks with a dip in between, two turning points.
Since f′ f' must have two turning points to account for two points of inflection in f f , Graph C or Graph D could represent f′ f' .
The graph that could represent f′ f' is:
Graph C or Graph D \boxed{\text{Graph C or Graph D}}

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