Concept explainers
The average value of the function f on [a,b] that lies between f(a) and f(b).
Answer to Problem 45E
It is not necessary that average value of a function f on [a, b] always lies between f(a) and f(b).
Explanation of Solution
Given information:
The average value of the function f on [a,b] that lies between f(a) and f(b).
Formula used:
Calculation:
Statement is false.
If f is integrated on [a, b], its average (mean) value on [a, b] is
If max f and min f are the maximum and minimum values of f on [a, b], then
Now, divide the inequality by (b - a)
So,
If function is monotonously increasing on [a, b]
So,
If function is monotonously decreasing on [a, b]
So,
But, if function is not monotonously increasing or decreasing on [a, b], then it is not necessary that average value of a function f on [a, b] always lies between f(a) and f(b).
Conclusion:
If the function is not monotonously increasing or decreasing on [a, b], then it is not necessary that average value of a function f on [a, b] always lies between f(a) and f(b).
Chapter 6 Solutions
Calculus: Graphical, Numerical, Algebraic
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