Math Problem Statement
Solution
The problem asks to find the points on the graph of the function where the tangent line is horizontal.
Steps to solve:
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Condition for a horizontal tangent line: A tangent line is horizontal where the derivative of the function is zero, i.e., .
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Find the derivative: The derivative of is:
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Set the derivative equal to zero: To find where the tangent line is horizontal, we solve : Simplifying:
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Find the corresponding points: Now we substitute and into the original function to find the points.
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For : So, one point is .
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For : So, the second point is .
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Conclusion:
The points where the tangent line is horizontal are and .
Would you like further details on this solution, or do you have any questions?
Here are some related questions:
- What is the significance of a horizontal tangent line in a graph?
- How does the derivative relate to the slope of the tangent line?
- What happens to the shape of the graph around the points where the tangent is horizontal?
- How would we determine if these points are local maxima or minima?
- Could there be any inflection points in this graph? If so, how would we find them?
Tip: Horizontal tangent lines typically occur at local maxima, minima, or saddle points where the slope of the curve is zero.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Critical Points
Formulas
Derivative of f(x) = 27x - x^3: f'(x) = 27 - 3x^2
Theorems
The point where the derivative is zero gives the horizontal tangent
Suitable Grade Level
Grades 11-12 (High School Calculus)
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