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JEE Main20252 Apr 2025Morning ShiftMathematicsDifferential EquationsActual

Let f: R R be a thrice differentiable odd function satisfying f^ ( x ) 0, f^ ( x )=f( x ), f(0)=0, f^ (0)=3 . Then 9 f ( _ c 3 ) is equal to _______.

Correct answer

0

Step-by-step solution

aligned & f ^ ( x )= f ( x ) & f ^ ( x ) f ^ ( x )= f ^ ( x ) f ( x ) & ( f ^ ( x ) )^2 2 = ( f ( x ))^2 2 + C & ( f ^ ( x ) )^2=( f ( x ))^2+ C ^ aligned f (0)=0, f ^ (0)=3 C ^ =9 ( f ^ ( x ) )^2=( f ( x ))^2+9 f ^ ( x )= ( f ( x ))^2+9 f ^ ( x ) 0 dy y ^2+9 = dx | y + y ^2+9 |= x + C f (0)=0 C = 3 y + y ^2+9 =3 e ^ x at x= 3 ; y=4 9 f ( 3)=36

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