Questions: The function f(x) is continuous on (-∞, ∞). Use the given information to sketch the graph of f. x -2 -1 0 1 2 f(x) -1 1.1 2 4 -4 Choose the correct graph of f below. A. B. C. D.

The function f(x) is continuous on (-∞, ∞). Use the given information to sketch the graph of f.

x -2 -1 0 1 2
f(x) -1 1.1 2 4 -4

Choose the correct graph of f below.
A.
B.
C.
D.

Transcript text: The function $f(x)$ is continuous on $(-\infty, \infty)$. Use the given information to sketch the graph of $f$. \begin{tabular}{c|rrrrr} x & -2 & -1 & 0 & 1 & 2 \\ \hline $\mathbf{f}(\mathrm{x})$ & -1 & 1.1 & 2 & 4 & -4 \end{tabular} Choose the correct graph of f below. A. B. C. D.

Solution

Solution Steps

Step 1: Analyze the given information

The function $ f(x) $ is continuous on $ (-\infty, \infty) $.
The derivative $ f'(x) $ changes signs at $ x = -1 $ and $ x = 1 $.
The values of $ f(x) $ at specific points are given in the table.

Step 2: Determine the behavior of $ f(x) $ based on $ f'(x) $

For $ x < -1 $, $ f'(x) > 0 $, so $ f(x) $ is increasing.
At $ x = -1 $, $ f'(x) = 0 $, indicating a possible local maximum or minimum.
For $ -1 < x < 0 $, $ f'(x) < 0 $, so $ f(x) $ is decreasing.
At $ x = 0 $, $ f'(x) = 0 $, indicating a possible local maximum or minimum.
For $ 0 < x < 1 $, $ f'(x) > 0 $, so $ f(x) $ is increasing.
At $ x = 1 $, $ f'(x) = 0 $, indicating a possible local maximum or minimum.
For $ x > 1 $, $ f'(x) < 0 $, so $ f(x) $ is decreasing.

Step 3: Plot the points and sketch the graph

Plot the points $(-2, -1)$, $(-1, 1.1)$, $ (0, 2) $, $ (1, 4) $, and $ (2, 4) $.
Connect the points considering the increasing and decreasing behavior determined in Step 2.

Final Answer

The correct graph of $ f(x) $ is option B.

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