Let f(x) = \(\int\limits_{a-1}^{a+1}e^{-(x-1)^2}dx\)
∴ f'(x) = 0 gives
-2(x - 1) \(e^{-(x-1)^2}\) = 0
⇒ x - 1 = 0 (∵ e-(x - 1)2 \(\neq0\))
⇒ x = 1
Now, f"(x) = \(e^{-(x-1)^2}\)(-2 + 4(x - 1)2)
f"(1) = e0(-2 + 0) = -2 < 0
∴ x = 1 is point of maxima
Hence, maximum value of \(\int\limits_{a-1}^{a+1}e^{-(x-1)^2}dx \) is
attained at x = 1