11finding h'(x) we get
f'(x)-2f(x)f'(x)+3f(x)2f'(x)
h'(x)=f'(x)(1-2f(x)+3f(x)2)
checking the D of the inner bracket we get
D<0
means f(x) is increasing
so f'(x)>0
hence h'(x) is >0
so h(x) is an increasing function
Let h(x)=f(x)-(f(x))2+(f(x))3 for every real no x.
Then-
(a)h is increasing whenever f in is increasing.
(b)h is increasing whenever f is decreasing.
(c)h is decreasing whenever f is decreasing.
(d)nothing can be said in general.
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2 Answers
111h(x)=f(x)-(f(x))2+(f(x))3
h'(x)=f'(x)-2f(x)f'(x)+3f(x)2f'(x)
=f'(x)[1-2f(x)+3f(x)2]
Discriminant of the bracket = 4-4x3
=-8
Since a>0 and D<0 for all values of x therefore the contents of the bracket is always positive
therefore
h'(x) = f'(x) x [+ve no]
therefore h'(x) is depends on the value of f'(x)
i.e. if f(x) is increasing(f(x)>0) then h'(x) is increasing
and similarly for decreasing............
therefore (a) and (c) is the ans
hope dis helps u........
