- #1
Inequality with the mean value theorem
- Thread starter icystrike
- Start date
In summary, the mean value theorem is a fundamental theorem in calculus that states the relationship between the slope of the tangent and secant lines of a function on a closed and open interval. It can be used to understand inequality, the relationship between average and instantaneous rate of change, and the concept of concavity. However, it can only be applied to functions that are continuous and differentiable.
- #2
icystrike
- 445
- 1
Help Please :)
I wonder if i got the right solution
- #3
- 9,568
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No, your last inequality is backwards (using 2).
- #4
icystrike
- 445
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LCKurtz said:
What do u think i should do?
- #5
icystrike
- 445
- 1
Help please :0
- #6
HallsofIvy
Science Advisor
Homework Helper
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I cannot see you first attachment so I have no idea what you did and cannot say what you did wrong.
- #7
What is the mean value theorem?
The mean value theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints.
How is the mean value theorem used to understand inequality?
The mean value theorem is used to prove inequalities by showing that the function is either increasing or decreasing over the interval. This allows us to compare the function values at the endpoints and at the point where the slopes of the tangent and secant lines are equal.
What does the mean value theorem tell us about the relationship between the average rate of change and instantaneous rate of change?
The mean value theorem states that the average rate of change of a function over an interval is equal to the instantaneous rate of change at some point within the interval. This means that the average rate of change can be used to approximate the instantaneous rate of change at a specific point.
Can the mean value theorem be applied to all functions?
No, the mean value theorem can only be applied to functions that are continuous on a closed interval and differentiable on an open interval. If a function does not meet these criteria, the mean value theorem cannot be used.
How does the mean value theorem relate to the concept of concavity?
The mean value theorem can be used to determine the concavity of a function by analyzing the sign of the difference between the average rate of change and instantaneous rate of change. If the difference is positive, the function is concave up, and if it is negative, the function is concave down. Additionally, the mean value theorem can be used to prove the existence of points of inflection on a function.
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