Questions: Suppose that f(x) is continuous on (-∞, ∞). Sketch the graph of f such that f(-2)=4, f(0)=0, f(2)=-4 f'(−2)=0, f'(0)=0, f'(2)=0 f'(x)>0 on (-∞,-2) and (2, ∞)

Transcript text: Suppose that $f(x)$ is continuous on $(-\infty, \infty)$. Sketch the graph of $f$ such that \[ \begin{array}{l} f(-2)=4, f(0)=0, f(2)=-4 \\ f^{\prime}(-2)=0, f^{\prime}(0)=0, f^{\prime}(2)=0 \end{array} \] \[ f^{\prime}(x)>0 \text { on }(-\infty,-2) \text { and }(2, \infty) \]

Solution

Solution Steps

Step 1: Identify the given points and their derivatives

The function $ f(x) $ is continuous and has the following points:

$ f(-2) = 4 $
$ f(0) = 0 $
$ f(2) = -4 $

The derivatives at these points are:

$ f^{\prime}(-2) = 0 $
$ f^{\prime}(0) = 0 $
$ f^{\prime}(2) = 0 $

Step 2: Determine the behavior of the function

The derivative $ f^{\prime}(x) $ is positive on the intervals:

$ (-\infty, -2) $
$ (2, \infty) $

This indicates that the function is increasing on these intervals.

Step 3: Analyze the critical points

The points where the derivative is zero, $ x = -2, 0, 2 $, are critical points. Since the derivative changes sign around these points, they are likely local maxima or minima.

Final Answer

The function $ f(x) $ has local maxima or minima at $ x = -2, 0, 2 $ and is increasing on $ (-\infty, -2) $ and $ (2, \infty) $.

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