MathDB
Problems
Contests
National and Regional Contests
India Contests
India LIMIT
2019 LIMIT
2019 LIMIT Category B
Problem 9
Problem 9
Part of
2019 LIMIT Category B
Problems
(2)
# of sols to trig equation
Source: LIMIT 2019 CBS1 P9
4/28/2021
The number of solutions of the equation
tan
x
+
sec
x
=
2
cos
x
\tan x+\sec x=2\cos x
tan
x
+
sec
x
=
2
cos
x
, where
0
≤
x
≤
π
0\le x\le\pi
0
≤
x
≤
π
, is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<span class='latex-bold'>(A)</span>~0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
<span class='latex-bold'>(B)</span>~1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
<span class='latex-bold'>(C)</span>~2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
<span class='latex-bold'>(D)</span>~3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
trigonometry
algebra
extrema of |x^2-1|
Source: LIMIT 2019 CBS2 P9
4/28/2021
Let
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
be given by
f
(
x
)
=
∣
x
2
−
1
∣
,
x
∈
R
f(x)=\left|x^2-1\right|,x\in\mathbb R
f
(
x
)
=
x
2
−
1
,
x
∈
R
Then
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
f
has local minima at
x
=
±
1
but no local maxima
<span class='latex-bold'>(A)</span>~f\text{ has local minima at }x=\pm1\text{ but no local maxima}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
f
has local minima at
x
=
±
1
but no local maxima
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
f
has a local maximum at
x
=
0
but no local minima
<span class='latex-bold'>(B)</span>~f\text{ has a local maximum at }x=0\text{ but no local minima}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
f
has a local maximum at
x
=
0
but no local minima
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
f
has local minima at
x
=
±
1
and a local maximum at
x
=
0
<span class='latex-bold'>(C)</span>~f\text{ has local minima at }x=\pm1\text{ and a local maximum at }x=0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
f
has local minima at
x
=
±
1
and a local maximum at
x
=
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
None of the above
<span class='latex-bold'>(D)</span>~\text{None of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
None of the above
algebra