Questions: Sketch a graph of y=f(x), such that f satisfies the following conditions: - The domain of f is (-∞, ∞) - The point (1,1) lies on the graph of y=f(x) - f'(x)=0 at x=-4 and x=1 f'(-4)=0 f'(1)=0 - f'(x)>0 on the interval (-∞,-4) - f'(x)<0 on the intervals (-4,1) and (1, ∞) - f''(x)=0 at x=-2 and x=1 f'(-2)=0 f''(1)=0 - f''(x)<0 on the intervals (-∞,-2) and (1, ∞) - f''(x)>0 on the interval (-2,1) Identify any local minima, local maxima, and inflection points.

Sketch a graph of y=f(x), such that f satisfies the following conditions:
- The domain of f is (-∞, ∞)
- The point (1,1) lies on the graph of y=f(x)
- f'(x)=0 at x=-4 and x=1 f'(-4)=0 f'(1)=0
- f'(x)>0 on the interval (-∞,-4)
- f'(x)<0 on the intervals (-4,1) and (1, ∞)
- f''(x)=0 at x=-2 and x=1 f'(-2)=0 f''(1)=0
- f''(x)<0 on the intervals (-∞,-2) and (1, ∞)
- f''(x)>0 on the interval (-2,1)

Identify any local minima, local maxima, and inflection points.

Transcript text: Sketch a graph of $y=f(x)$, such that $f$ satisfies the following conditions: - The domain of $f$ is $(-\infty, \infty)$ - The point $(1,1)$ lies on the graph of $y=f(x)$ - $f^{\prime}(x)=0$ at $x=-4$ and $x=1 \quad f^{\prime}(-4)=0 \quad f^{\prime}(1)=0$ - $f^{\prime}(x)>0$ on the interval $(-\infty,-4)$ - $f^{\prime}(x)<0$ on the intervals $(-4,1)$ and $(1, \infty)$ - $f^{\prime \prime}(x)=0$ at $x=-2$ and $x=1 \quad f^{\prime}(-2)=0 \quad f^{\prime \prime}(1)=0$ - $f^{\prime \prime}(x)<0$ on the intervals $(-\infty,-2)$ and $(1, \infty)$ - $f^{\prime \prime}(x)>0$ on the interval $(-2,1)$ Identify any local minima, local maxima, and inflection points.

Solution

Solution Steps

Step 1: Analyze the given conditions

The function $ f(x) $ has the following characteristics:

The domain is $ (-\infty, \infty) $.
The point $ (1, 1) $ is on the graph.
Critical points where $ f'(x) = 0 $ are at $ x = -4 $ and $ x = 1 $.
$ f'(x) > 0 $ on $ (-\infty, -4) $.
$ f'(x) < 0 $ on $ (-4, 1) $ and $ (1, \infty) $.
Inflection points where $ f''(x) = 0 $ are at $ x = -2 $ and $ x = 1 $.
$ f''(x) < 0 $ on $ (-\infty, -2) $ and $ (1, \infty) $.
$ f''(x) > 0 $ on $ (-2, 1) $.

Step 2: Determine local minima and maxima

Since $ f'(x) $ changes from positive to negative at $ x = -4 $, there is a local maximum at $ x = -4 $.
Since $ f'(x) $ changes from negative to negative at $ x = 1 $, there is no local extremum at $ x = 1 $.

Step 3: Determine inflection points

Since $ f''(x) $ changes from negative to positive at $ x = -2 $, there is an inflection point at $ x = -2 $.
Since $ f''(x) $ changes from positive to negative at $ x = 1 $, there is an inflection point at $ x = 1 $.

Final Answer

Local maximum at $ x = -4 $.
Inflection points at $ x = -2 $ and $ x = 1 $.

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