48. Sketch the graph of a function $$ f $$ that has all of the following properties: a. The graph has discontinuities at $$ x=-1 $$ and at $$ x=3 $$ b. $$ f^{\prime}(x)0 $$ for $$ x0 $$ for $$ x3 $$ and $$ f^{\prime \prime}(x)

Question

48. Sketch the graph of a function $$ f $$ that has all of the following properties: a. The graph has discontinuities at $$ x=-1 $$ and at $$ x=3 $$ b. $$ f^{\prime}(x)>0 $$ for $$ x<1, x eq-1 $$ c. $$ f^{\prime}(x)<0 $$ for $$ x>1, x eq 3 $$ d. $$ f^{\prime \prime}(x)>0 $$ for $$ x<-1 $$ and $$ x>3 $$ and $$ f^{\prime \prime}(x)<0 $$ for $$ -1

UpStudy AI · Accepted Answer

To sketch the graph of the function $$ f $$ with the given properties, we will analyze each condition step by step: ### Step 1: Identify Discontinuities - The graph has discontinuities at $$ x = -1 $$ and $$ x = 3 $$. This means the function is not defined at these points. ### Step 2: Analyze the First Derivative $$ f' $$ - **For $$ x < 1 $$ (excluding $$ x = -1 $$)**: $$ f'(x) > 0 $$ indicates that the function is increasing. - **For $$ x > 1 $$ (excluding $$ x = 3 $$)**: $$ f'(x) < 0 $$ indicates that the function is decreasing. ### Step 3: Analyze the Second Derivative $$ f'' $$ - **For $$ x < -1 $$**: $$ f''(x) > 0 $$ indicates that the function is concave up. - **For $$ -1 < x < 3 $$**: $$ f''(x) < 0 $$ indicates that the function is concave down. - **For $$ x > 3 $$**: $$ f''(x) > 0 $$ indicates that the function is concave up again. ### Step 4: Specific Function Values - $$ f(0) = 0 $$ - $$ f(2) = 0 $$ - $$ f(1) = 3 $$ ### Step 5: Sketching the Graph 1. **From $$ -\infty $$ to $$ -1 $$**: The function is increasing and concave up. It approaches the discontinuity at $$ x = -1 $$. 2. **At $$ x = -1 $$**: There is a discontinuity, so the graph jumps or is undefined. 3. **From $$ -1 $$ to $$ 1 $$**: The function is decreasing and concave down. It must pass through $$ (0, 0) $$ and reach a maximum at $$ (1, 3) $$. 4. **From $$ 1 $$ to $$ 3 $$**: The function is decreasing and concave down, and it must pass through $$ (2, 0) $$ before reaching the discontinuity at $$ x = 3 $$. 5. **At $$ x = 3 $$**: There is another discontinuity. 6. **From $$ 3 $$ to $$ \infty $$**: The function is increasing and concave up. ### Final Sketch The graph will have the following characteristics: - It will increase from the left towards $$ x = -1 $$, then jump or drop at $$ x = -1 $$. - It will decrease from $$ x = -1 $$ to $$ x = 1 $$, reaching a maximum at $$ (1, 3) $$. - It will decrease again from $$ (1, 3) $$ to $$ (2, 0) $$ and then drop at $$ x = 3 $$. - Finally, it will increase from $$ x = 3 $$ onwards. This sketch captures all the properties of the function $$ f $$. To sketch the graph of function $$ f $$, follow these steps: 1. **Discontinuities**: Draw vertical breaks at $$ x = -1 $$ and $$ x = 3 $$. 2. **Increasing and Decreasing**: - For $$ x < 1 $$ (excluding $$ x = -1 $$), the graph slopes upward. - For $$ x > 1 $$ (excluding $$ x = 3 $$), the graph slopes downward. 3. **Concavity**: - For $$ x < -1 $$ and $$ x > 3 $$, the graph is concave up. - For $$ -1 < x < 3 $$, the graph is concave down. 4. **Specific Points**: - $$ f(0) = 0 $$ - $$ f(2) = 0 $$ - $$ f(1) = 3 $$ The graph will have an increasing trend before $$ x = -1 $$, a peak at $$ x = 1 $$, a decrease to $$ x = 2 $$, and then an increasing trend after $$ x = 3 $$, with discontinuities at $$ x = -1 $$ and $$ x = 3 $$.

Sketch the graph of a function that has all of the
following properties:
a. The graph has discontinuities at and at

b. for
c. for
d. for and and
for
e.

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Solución

Step 1: Identify Discontinuities

Step 2: Analyze the First Derivative

Step 3: Analyze the Second Derivative

Step 4: Specific Function Values

Step 5: Sketching the Graph

Final Sketch

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Sketch the graph of a function that has all of the following properties: a. The graph has discontinuities at and at b. for c. for d. for and and for e.