Math Problem Statement
Use the graph of the function f(x) = x3 + x2 − x − 1 to identify its relative maximum and minimum.
Solution
To identify the relative maximum and minimum of the function using the graph:
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Relative Maximum: This occurs at the highest point of a turning region on the graph. From the image, it appears there is no clear peak above the x-axis, indicating there is no relative maximum.
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Relative Minimum: The graph shows a clear turning point around where the graph dips below the x-axis before rising again. This point, based on the shape, would be the relative minimum.
Thus, the relative minimum occurs near , where reaches its lowest point locally.
Would you like more details or further clarification?
Relative Questions:
- How do we calculate exact relative extrema using calculus?
- What is the derivative of the given function?
- How do we distinguish between relative maximum and minimum using the second derivative test?
- Can the function have absolute extrema if it is a cubic polynomial?
- How does the shape of cubic functions generally behave in terms of turning points?
Tip:
When identifying relative extrema graphically, look for points where the slope of the function changes sign, indicating a transition from increasing to decreasing or vice versa.
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Math Problem Analysis
Mathematical Concepts
Calculus
Graph Analysis
Cubic Functions
Formulas
f(x) = x^3 + x^2 − x − 1
First Derivative: f'(x) = 3x^2 + 2x − 1
Second Derivative: f''(x) = 6x + 2
Theorems
Relative Maximum and Minimum (First Derivative Test)
Second Derivative Test
Suitable Grade Level
Grades 10-12
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