MATH INTEGRATION

Question

The maximum value of \(\int\limits_{a-1}^{a+1}e^{-(x-1)^2}dx\) is attained (a is real ) at

Answer 1

the answer of the above qiestion is in the picture

Answer 2

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