JEE Advanced2012MathematicsApplication of DerivativesActual

If f(x)= ₀^ x e^ t² (t-2)(t-3) d t for all x (0, ) , then

Options

Af has a local maximum at x=2
Bf is decreasing on (2,3)
Cthere exists some c (0, ) , such that f^ (c)=0
Df has a local minimum at x=3

Correct answer

A. f has a local maximum at x=2

Step-by-step solution

f(x)= ₀^ x e^ t² (t-2)(t-3) d t f^ (x)=e^ x² (x-2)(x-3) Put f^ (x)=0 x=2,3f^ (x)=e^ x² 2 x (x²-5 x+6 )+e^ x² (2 x-5)f^ (2)=- ve and f^ (3)=+ ve x=2 is a point of local maxima. and x=3 is a point of local minima. Also for x (2,3), f^ (x) 0 There exists some C (0,1) such that f^ (C)=0 Hence all the options are correct.

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