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Byju's Answer
Standard X
Mathematics
Polynomial
If f(X)=0 is ...
Question
If f(X)=0 is a polynomial and f(a),f(b) have opposite sign,then f(X)=0 has atleast one real root in the interval (a,b) Prove the statement
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Similar questions
Q.
Assertion :If
0
<
α
<
(
π
/
4
)
, then the equation
(
x
−
s
i
n
α
)
(
x
−
c
o
s
α
)
−
2
=
0
has both roots in
(
s
i
n
α
,
c
o
s
α
)
. Reason: If
f
(
a
)
and
f
(
b
)
possess opposite signs, then there exists at least one solution of the equation
f
(
x
)
=
0
in open interval
(
a
,
b
)
.
Q.
Assertion (A) : One root of
x
3
−
2
x
2
−
1
=
0
lies between 2 and 3.
Reason (R) : If
f
(
x
)
is continuous function and
f
(
a
)
,
f
(
b
)
have opposite sings then atleast one root of
f
(
x
)
=
0
lies between
a
and
b
.
Q.
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
−
3
be a polynomial with real coefficient. Let
x
1
,
x
2
,
x
3
,
(
x
i
>
0
)
be the roots of
f
(
x
)
=
0
, such that
3
∑
i
,
j
,
k
=
1
i
≠
j
≠
k
x
i
x
j
+
x
k
=
3
2
. Then which of following(s) is/are possible ?
Q.
Assertion :Statement-1 : If
a
,
b
in
R
and
a
<
b
, then there is atleast one real number
c
∈
(
a
,
b
)
such that
c
a
+
b
=
b
2
+
a
2
4
c
2
.
Reason: Statement-2 : If
f
(
x
)
is continuous in
[
a
,
b
]
and derivable in
(
a
,
b
)
&
f
′
(
c
)
=
0
for atleast one
c
∈
(
a
,
b
)
, then it necessarily implies that
f
(
a
)
=
f
(
b
)
.
Q.
Assertion :If
27
a
+
9
b
+
3
c
+
d
=
0
, then the equation
f
(
x
)
=
4
a
x
3
+
3
b
x
2
+
2
c
x
+
d
=
0
has at least one real root lying between
(
0
,
3
)
. Reason: If
f
(
x
)
is continuous in
[
a
,
b
]
, derivable in
(
a
,
b
)
such that
f
(
a
)
=
f
(
b
)
, then there exists at least one point
c
∈
(
a
,
b
)
such that
f
′
(
c
)
=
0
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