JEE Main20253 Apr 2025Morning ShiftMathematicsContinuity and DifferentiabilityActual

Let f( x )= cases (1+ ax )^ 1 / x & , x 0 1+ b & , x =0 ( x +4)^ 1 / 2 -2 ( x + c )^ 1 / 3 -2 & , cases be continuous at x=0 . Then e^a b c is equal to

Options

Correct answer

C. 48

Step-by-step solution

aligned & f (0⁻ )= e ^ _ x 0 ax x = e ^ a & f (0)=1+ b & f (0⁺ )= 1 2 x +4 1 3 ( x + c )^ - 2 3 = 1 2(2) 1 3 c ^ - 2 3 & = 3 4 c ^ 2 / 3 aligned Also at x =0 ; c^ 1 / 3 =2 c=8 So f (0⁺ )= 3 4 (8)^ 2 / 3 =3 Now, e ^ a = b +1=3 e ^ a . b c =3 2 8=48

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