csat-suneung 2017 Q20

csat-suneung · South-Korea · csat__math-science 4 marks Differentiating Transcendental Functions Differentiation under the integral sign with transcendental kernels

For the function $f ( x ) = e ^ { - x } \int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$, which of the following statements are correct? [4 points]
ㄱ. $f ( \sqrt { \pi } ) > 0$ ㄴ. There exists at least one $a$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( a ) > 0$. ㄷ. There exists at least one $b$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( b ) = 0$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the function $f ( x ) = e ^ { - x } \int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$, which of the following statements are correct? [4 points]

ㄱ. $f ( \sqrt { \pi } ) > 0$\\
ㄴ. There exists at least one $a$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( a ) > 0$.\\
ㄷ. There exists at least one $b$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( b ) = 0$.\\
(1) ㄱ\\
(2) ㄷ\\
(3) ㄱ, ㄴ\\
(4) ㄴ, ㄷ\\
(5) ㄱ, ㄴ, ㄷ

Paper Questions

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