Question

Q67. If f(x) is continuous in (a, b) and if f(a)
and f(b) are of opposite signs then the
equation f(x) = 0 will have at least
be ween 'a' and 'b'.​

Answer 1

Given :   f(x) is continuous in (a, b)   f(a)  and f(b) are of opposite signs

then the  equation f(x) = 0 will have at least    ____ between 'a' and 'b'.​

To Find : Fill in the Blank

Solution:

f(x) is continuous in (a, b)

 f(a)  and f(b) are of opposite signs

Case 1

f(a) > 0 (+)  and  f(b) < 0  (-)

Then function is decreasing from f(a) to f(b)

and will intersect x axis between a & b

Hence will have atleast one root between a & b

Case 1

f(a) < 0  (- ) and  f(b) > 0  ( + )

Then function is increasing from f(a) to f(b)

and will intersect x axis between a & b

Hence will have atleast one root between a & b

equation f(x) = 0 will have at least   one root / Solution____ between 'a' and 'b'.​

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