increasing/decreasing problem

[chirag]

Let f and g be two functions defiNed on an interval I such that f(x)>=0 and g(x)<=0 for all x belongs to I, and f is strictly decreasing on I while g is strictly increasing on I. Then
A) the product function fg is strictly increasing on I;
B) the product function fg is strictly decreasing on I;
C) the product function fg is increasing but not necessraily strictly increasing on I;
D) nothing can be said about the monotonicity of the product function fg.

[Abhitesh]

Let h = fg
=> h' = f'g + g'f
           (-)(<=0)  +   (+)(>=0)
           (>=0)   + (>=0)
           (>=0)

Answer should be (C)

[chirag]

Answer is option A not C.

[RichaS]

For it to be strictly increasing it has to >0 not >=0

[Abhitesh]

One correction :-

Given the conditions f and g cannot be zero at the same point.
So h'>0. Hence option A.

[RichaS]

Ummm why can't f and g be 0 at the same point?

[chirag]

Yes same doubt what Richa has.

[Abhitesh]

The conditions are -
1. f and g are defined over entire interval I. Lets say I=[a,b]. It can (a,b) or any combination but closed interval simplifies the explanation.
2. f>=0 and g<=0 for all x in I.
3. f is strictly decreasing and g is strictly increasing.

Now suppose that both f and g are 0 at c; a<c<b.
Then from condition (3) f(x)<0 and g(x)>0 for c<x<b which violates condition (2)

Similar explanation when f=g=0 at a

If at all only f(b)=0 and g(b)=0. So h'(b) can be zero in case of closed interval. Seems option C is correct.
Let me know if there is alternative explanation.