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JEE Advanced2011MathematicsDefinite IntegrationActual

Let f:[1, ) [2, ) be a differentiable function such that f(1)=2 . If 6 ₁^x f(t) d t=3 x f(x)-x^3 for all x 1 , then the value of f(2) is

Correct answer

0

Step-by-step solution

Given, f(1)= 1 3 and 6 ₁^x f(t) d t=3 x f(x)-x^3 , for all x 1 Using (Newton-Leibnitz formula), On Differentiating both sides, aligned & 6 f(x) 1-0-3 f(x)+3 x f^ (x)-3 x^2 & 3 x f^ (x)-3 f(x)=3 x^2 & f^ (x)- 1 x f(x)=x aligned x f^ (x)-f(x) x^2 =1 d d x f(x) x =1 On integrating both sides, array ll & f(x) x =x+C [ f(1)= 1 3 ] & 1 3 =1+C C=- 2 3 & Now, f(x)=x^2- 2 3 x & f(2)=4- 4 3 = 8 3 array Note Here, f(1)=2 does not satisfy given function. f(1)= 1 3 For that, f(x)=x^2- 2 3 x and f(2)=4- 4 3 = 8 3

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