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Joint Entrance Examination
Mathematics
Application of Derivatives
Question 193
Question 193 of 212
Medium
MCQ
Let the function \[f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6\] have a maxima for some value of
x
0
x 0
x
0
. Then, the set of all values of p is
A
\[\left( { - {9 \over 2},{9 \over 2}} \right)\]
B
\[\left( {{9 \over 2},\infty } \right)\]
C
\[\left( {0,{9 \over 2}} \right)\]
D
\[\left( { - \infty ,{9 \over 2}} \right)\]
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Official solution
f
′
(
x
)
=
6
x
2
+
2
x
(
2
p
−
7
)
+
3
(
2
p
−
9
)
f^{\prime}(x)=6 x^{2}+2 x(2 p-7)+3(2 p-9)
f
′
(
x
)
=
6
x
2
+
2
x
(
2
p
−
7
)
+
3
(
2
p
−
9
)
x
1
0
x_{1}0
x
1
0
⇒
f
′
(
0
)
<
0
\Rightarrow f^{\prime}(0)<0
⇒
f
′
(
0
)
<
0
⇒
p
<
9
2
\Rightarrow p<\frac{9}{2}
⇒
p
<
2
9
\[p \in \left( { - \infty ,{9 \over 2}} \right)\]
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