Find the formula for a function of the form $C(x)=e^{-(x-a)^{2} / b}$ for $b> 0$ with (i) a local maximum at $x=-1$ and (ii) points of inflection at $x=-5$ and $x=3$
\[
C(x)=
\]
Check your result by making a plot (not submitted) .

Step 1 ：Find the first and second derivatives of C(x):

Step 2 ：\[C'(x) = -(-2a + 2x)e^{-(x-a)^{2}/b}/b\]

Step 3 ：\[C''(x) = -2e^{-(x-a)^{2}/b}/b + (-2a + 2x)^{2}e^{-(x-a)^{2}/b}/b^{2}\]

Step 4 ：Use the given conditions to find the values of a and b:

Step 5 ：For a local maximum at x = -1, set C'(x) = 0:

Step 6 ：\[0 = -(-2a - 2)e^{-(1+a)^{2}/b}/b\]

Step 7 ：Solve for a: a = -1

Step 8 ：For points of inflection at x = -5 and x = 3, set C''(x) = 0:

Step 9 ：\[0 = -2e^{-(5+a)^{2}/b}/b + (-2a - 10)^{2}e^{-(5+a)^{2}/b}/b^{2}\]

Step 10 ：\[0 = -2e^{-(3-a)^{2}/b}/b + (6 - 2a)^{2}e^{-(3-a)^{2}/b}/b^{2}\]

Step 11 ：Solve for b: b = 2a^{2} - 12a + 18

Step 12 ：Substitute the value of a into the equation for b:

Step 13 ：b = 2(-1)^{2} - 12(-1) + 18 = 30

Step 14 ：\boxed{C(x) = e^{-(x+1)^{2}/30}}