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A twice–differentiable function satisfies for all real
MATHEMATICS · Single correct · CBSE · Mathematics · Integrals · Integrals
Problem
A twice–differentiable function
g
g
g
satisfies
g
′
′
(
x
)
=
e
2
x
−
4
g''(x)=e^{2x}-4
g
′′
(
x
)
=
e
2
x
−
4
for all real
x
x
x
. If
g
′
(
0
)
=
3
g'(0)=3
g
′
(
0
)
=
3
and
g
(
1
)
=
5
g(1)=5
g
(
1
)
=
5
, which of the following is
g
(
x
)
g(x)
g
(
x
)
? (a)
1
2
e
2
x
−
2
x
2
+
2
x
+
5
−
1
2
e
2
\dfrac12 e^{2x}-2x^{2}+2x+5-\dfrac12 e^{2}
2
1
e
2
x
−
2
x
2
+
2
x
+
5
−
2
1
e
2
(b)
1
4
e
2
x
−
2
x
2
+
5
2
x
+
9
2
−
1
4
e
2
\dfrac14 e^{2x}-2x^{2}+\dfrac52 x+\dfrac92-\dfrac14 e^{2}
4
1
e
2
x
−
2
x
2
+
2
5
x
+
2
9
−
4
1
e
2
(c)
1
4
e
2
x
−
2
x
2
+
5
2
x
+
9
2
+
1
4
e
2
\dfrac14 e^{2x}-2x^{2}+\dfrac52 x+\dfrac92+\dfrac14 e^{2}
4
1
e
2
x
−
2
x
2
+
2
5
x
+
2
9
+
4
1
e
2
(d)
1
4
e
2
x
+
5
2
x
+
5
2
−
1
4
e
2
\dfrac14 e^{2x}+\dfrac52 x+\dfrac52-\dfrac14 e^{2}
4
1
e
2
x
+
2
5
x
+
2
5
−
4
1
e
2
Related theory
Higher-Order Derivatives
Advanced Differentiation
Antiderivatives
Standard algebraic integrals
Integration as inverse process of differentiation
Integrals
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