User:Egm6341.s11.team4.KTH/HW1

Find
1-4 a) Prove IMVT P.4-1 for nonnegative function, $$ w(\cdot ) $$. i.e., $$ w(\cdot)\geq 0,  \forall x \in [a,b]  $$

1-4 b) Another version of IMVT $$ w(\cdot ) \neq 0, \forall x \in [a,b] $$. Either $$ w(\cdot)>0 $$ or $$ w(\cdot)<0 $$

Given
Continuous function

Solution
1-4 a)

Integrate both sides over closed interval a,b.

Let,

According to Intermediate Value Theorem

1-4 b)

Integrate both sides over closed interval a,b.

Let,

According to Intermediate Value Theorem

From 4-1 a) From 4-1 b)