Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must int 0^2 f(x)dx lie? Which property of integrals allows you to make your conclusion?

Wotzdorfg

Wotzdorfg

Answered question

2020-10-18

Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must 02f(x)dx lie? Which property of integrals allows you to make your conclusion?

Answer & Explanation

sweererlirumeX

sweererlirumeX

Skilled2020-10-19Added 91 answers

Step 1
Given that f(x) has an absolute maximum of M and an absolute minimum of m
So, m<f(x)<M
Next, we integrate all sides between 0 and 2.
m<f(x)<M
02m dx <02f(x) dx <02M dx 
m02 dx <02f(x) dx <M02 dx 
m[x]02<02f(x) dx <M[x]02
m(20)<02f(x) dx <M(20)
2m<02f(x) dx <2M
Step 2
Using the extreme value theorem and comparison property of integrals we made our conclusion comparison property of integrals:
 if mf(x)Mf or axb,then
m(ba)abf(x) dx M(ba)

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