Let f be a differentiable function satisfying f(x)=1-2 x+ _ 0 ^ x e ^ (x-t) f(t) dt , x R and let g (x)= _ 0 ^ x (f( t )+2)^ 15 ( t -4)^ 6 ( t +12)^ 17 dt , x R . If p and q are respectively the points of local minima and local maxima of g, then the value of | p + q | is equal to \_\_\_\_.
A. A
Given f(x) = 1 - 2x + _0^x e^ (x-t) f(t) \, dt. Differentiating and simplifying: e^ -x f'(x) - e^ -x f(x) = -2e^ -x + (1-2x)e^ -x (-1) + e^ -x f(x) This reduces to: f'(x) - 2f(x) = 2x - 3 Solving the linear ODE dy dx - 2y = 2x - 3: y e^ -2x = e^ -2x (2x-3)\,dx On solving, we get f(x) = 1 - x. Now, g(x) = _0^x (3-t)^ 15 (t-4)^6(t+12)^ 17 \, dt g'(x) = (3-x)^ 15 (x-4)^6(x+12)^ 17 = -(x-3)^ 15 (x-4)^6(x+12)^ 17 Sign analysis of g'(x): g'(x) changes from + - at x = 3 → local maxima (q = 3) g'(x) changes from - + at x = -12 → local minima (p = -12) |p + q| = |-12 + 3| = 9
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