Thread: need help integral theorems

State true and false
1>If f and g are contineous and f(x)>= g(x) for a<x<b, then integral(a to b)f(x)dx>=integral(a tob)g(x)dx
2>If f and g are contineous and f(x)>= g(x) for a<x<b, then f '(x)>=g'(x) for a<x<b

#1 is a theorem. You should see if you could prove it. It is actually very important.


#2: On [0,1] consider

Hello, Gracy!

State true and false.

(1) If and are continuous and for ,
then: .

I would say True . . .

Code:

        |   *
        |   :*        :
        |   :  *      :
        |   :     *   :
        |   :         * f(x)
        |   :         :
        |   :         o g(x)
        |   :      o  :
        |   :  o      :
        |   o         :
      --+---+---------+--
        |   a         b

If is always "above" on
. . then the area under will be the area under

(2) If and are continuous and for ,
then: . for

This is False . . .

Code:

        |   *
        |   :*        :
        |   :  *      :
        |   :     *   :
        |   :         * f(x)
        |   :         :
        |   :         o g(x)
        |   :      o  :
        |   :  o      :
        |   o         :
      --+---+---------+--
        |   a         b

While is above , we see that
. . is negative while is positive.