Math  /  Calculus

QuestionQuestion The twice-differentiable function ff is shown below on the domain (9,9)(-9,9). The function ff has points of inflection at x=7.6x=-7.6, x=1.3,x=2.4x=-1.3, x=2.4, shown with small green circles on the graph. Determine what could be said about the values of f(3),f(3)f(3), f^{\prime}(3), and f(3)f^{\prime \prime}(3).
Answer Attempt 1 out of 2 f(3) \qquad and f(3)f^{\prime}(3) \square and f(3)f^{\prime \prime}(3) \square Submit Answer

Studdy Solution

STEP 1

What is this asking? Figure out what the function f f , its first derivative f(3) f'(3) , and its second derivative f(3) f''(3) are doing at x=3 x = 3 . Watch out! Don't confuse points of inflection with local maxima or minima.
Also, remember that the second derivative tells us about concavity!

STEP 2

1. Analyze the function f(3) f(3)
2. Determine the behavior of f(3) f'(3)
3. Understand the concavity using f(3) f''(3)

STEP 3

Let's start by looking at the graph to see what's happening at x=3 x = 3 .
The graph shows that the function is increasing at this point.
This means that the value of f(3) f(3) is somewhere above where it was just before x=3 x = 3 .
We can't give an exact number without more information, but we can say that the function is **increasing**.

STEP 4

Since the graph is increasing at x=3 x = 3 , the first derivative f(3) f'(3) must be **positive**.
Why? Because the first derivative tells us the slope of the tangent line to the curve at that point.
If the slope is positive, the function is increasing!

STEP 5

Now, let's talk about the concavity at x=3 x = 3 .
The second derivative f(3) f''(3) tells us about the concavity.
Since x=3 x = 3 is after the last point of inflection at x=2.4 x = 2.4 , and the graph is still increasing, we need to determine if it's concave up or down.

STEP 6

Look at the graph after the point of inflection at x=2.4 x = 2.4 .
If the graph is curving upwards, f(3) f''(3) is **positive** (concave up).
If it's curving downwards, f(3) f''(3) is **negative** (concave down).
Without the graph, we can't be sure, but we can say that the concavity changes at inflection points, so check the graph carefully!

STEP 7

At x=3 x = 3 , the function f f is increasing, so f(3)>0 f'(3) > 0 .
The exact value of f(3) f(3) isn't given, but it's increasing.
The concavity f(3) f''(3) depends on the graph's curve after the last point of inflection at x=2.4 x = 2.4 .

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