Math Problem Statement
Find all the critical numbers of f left parenthesis x right parenthesis equals cube root of 2 x squared minus 18 end root, then determine the local minimum and maximum points by using a graph.
Solution
To find the critical numbers of the function , we need to follow these steps:
- Find the first derivative of .
- Set the first derivative equal to zero and solve for to find critical points.
- Check where the first derivative is undefined.
- Determine the local minimum and maximum points using a graph.
Step 1: Find the First Derivative
The function is .
Using the chain rule for differentiation:
Let . Then .
Using the chain rule:
Step 2: Set the First Derivative Equal to Zero and Solve for
This equation is zero when :
Step 3: Check Where the First Derivative is Undefined
The first derivative is undefined when the denominator is zero: This occurs when:
Critical Numbers
The critical numbers are and .
Step 4: Determine Local Minimum and Maximum Points Using a Graph
We need to analyze the behavior of at these critical points using the first and second derivative tests, but we can also use a graph to visually determine the local minimum and maximum points.
Plotting :
By graphing, we can observe the behavior of the function near the critical points.
Critical Point Analysis
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At :
- The function has an inflection point here since the graph changes concavity but does not achieve a local minimum or maximum.
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At :
- The function achieves a local minimum here.
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At :
- The function achieves a local maximum here.
Conclusion
- Critical numbers:
- Local minimum point:
- Local maximum point:
Would you like further details or have any questions?
Here are five questions you might ask:
- How do you perform the first and second derivative tests?
- Can you explain how to graph a function using critical points?
- What is the significance of an inflection point?
- How do you determine if a function is increasing or decreasing?
- What are some common mistakes to avoid when finding critical points?
Tip: Always check where the derivative is undefined as these points can be critical for identifying the behavior of the function.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Critical Points
Graphical Analysis
Formulas
Chain Rule
First Derivative Test
Second Derivative Test
Theorems
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Suitable Grade Level
Advanced High School / College