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Question

Let f,g:[2,3]R be continuous functions which are twice differentiable on the interval (2,3). Let the values of f and g at the points 2, 0, 1 and 3 be as given in table

x=2 x=0 x=1 x=3
f(x) 11 -4 6 16
g(x) 1 -2 0 2

In each of the intervals (2,0), (0,1) and
(1,3) the function (f5g)′′ never vanishes. Then the correct statement(s) is/are

A
f(x)5g(x)=0 has exactly three solutions in(2,0)(0,1)(1,3)
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B
f(x)5g(x)=0 has exactly one solution in (0,1)
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C
f(x)5g(x)=0 has atleast one solution in (1,3)
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D
f(x)5g(x)=0 has atleast two solutions in (2,0)
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Solution

The correct option is C f(x)5g(x)=0 has atleast one solution in (1,3)
Let h(x)=f(x)5g(x)
h(x) is continuous, differentiable and f(x) is also continuous, differentiable
h(2)=h(0)=h(3)=h(1)=6
and (f5g)′′ never vanishes
h′′(x)0
h(x) never changes its nature in (2,0),(0,1),(1,3).
So using rolle's theorem there exists exactly one solution for h(x)=0 in each of the three intervals (2,0),(0,1) and (1,3).

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