jee-advanced 2005 Q17

jee-advanced · India · mains Proof Existence or properties of extrema via abstract/theoretical argument
17. $f ( x )$ is a differentiable function and $g ( x )$ is a double differentiable function such that $| f ( x ) | < 1$ and $f ^ { \prime } ( x ) = g ( x )$. If $f ^ { 2 } ( 0 ) + g ^ { 2 } ( 0 ) = 0$. Prove that there exists some $c \hat { I } ( - 3,3 )$ such that $\mathrm { g } ( \mathrm { c } ) . \mathrm { gn } ( \mathrm { c } ) < 0$.
If $y = y ( x )$ satisfies the relation $x \cos y + y \cos x = \pi$, then $y ^ { \prime \prime } ( 0 )$ equals 
17. $f ( x )$ is a differentiable function and $g ( x )$ is a double differentiable function such that $| f ( x ) | < 1$ and $f ^ { \prime } ( x ) = g ( x )$. If $f ^ { 2 } ( 0 ) + g ^ { 2 } ( 0 ) = 0$. Prove that there exists some $c \hat { I } ( - 3,3 )$ such that $\mathrm { g } ( \mathrm { c } ) . \mathrm { gn } ( \mathrm { c } ) < 0$.\\
Paper Questions
Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18