Math Problem Statement
Show that the function f: [1,4]→R defined by f(x) = x ^ 3 - 3x is an increasing function
Solution
To show that the function is an increasing function on the interval , we need to examine its derivative, as the derivative of a function tells us about its rate of change.
Step 1: Find the derivative of
The derivative of is:
Step 2: Analyze the sign of
Now, we need to determine whether for all . This will tell us if is increasing throughout the interval.
Now, let's analyze the sign of on the interval .
- At , .
- For , since and , .
Thus, for , , meaning the function is non-decreasing, and since for , it is strictly increasing on .
Conclusion:
The function is increasing on the interval because its derivative is non-negative and positive for .
Would you like more details or have any questions?
Here are 5 related questions you might want to explore:
- What is the difference between a strictly increasing and a non-decreasing function?
- How does the second derivative of a function help in determining concavity?
- Can a function be increasing even if its derivative is zero at some points?
- How do we classify critical points in a function like ?
- What is the importance of the Mean Value Theorem in understanding function behavior?
Tip: Always check the derivative of a function over a specific interval to determine whether the function is increasing, decreasing, or constant.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Increasing and Decreasing Functions
Formulas
f'(x) = 3x^2 - 3
f'(x) = 3(x - 1)(x + 1)
Theorems
First Derivative Test
Suitable Grade Level
Grades 11-12
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