Find a formula for a curve of the form y=e^((−(x−a)^2)/b) for b greater than 0 with a local maximum at x=−5 and points of inflection at x=−8 and x=−2.

To find a formula for a curve in the form y = e^((-((x-a)^2)/b)), we need to determine the values of 'a' and 'b' that satisfy the given conditions.

1. Local maximum at x = -5:
For a local maximum, the first derivative of the curve should be zero at x = -5. Let's first find the first derivative:

y = e^((-((x-a)^2)/b))
Take the natural logarithm (ln) of both sides:
ln(y) = (-((x-a)^2)/b)

Now, we can differentiate both sides with respect to x:
(d/dx) ln(y) = (d/dx) (-((x-a)^2)/b)
(1/y) (dy/dx) = -2(x-a)/(b)

Since we want a local maximum at x = -5, let's substitute these values into our equation and find the value of 'a':
(-2(-5-a))/(b*y) = 0

Simplifying the equation, we get:
10 + 2a = 0
2a = -10
a = -5

So, we have found the value of 'a' for the local maximum condition.

2. Points of inflection at x = -8 and x = -2:
For points of inflection, the second derivative of the curve should be zero at the given x-values. Let's find the second derivative of our curve:

(d^2/dx^2) ln(y) = (d/dx) (-2(x-a)/(b))
(1/y) (d^2y/dx^2) - (dy/dx)^2/y^2 = -2/b

Now, let's substitute x = -8 and solve for 'b':
(-2/b)*y = 0
-2/b = 0
This equation cannot hold true, which means that a curve of the given form does not have a point of inflection at x = -8. We need to reconsider our curve equation.

The correct form of the curve for the given conditions is:
y = e^((-((x+5)^2)/b))

Now, let's find the value of 'b' for the point of inflection at x = -2.

For a point of inflection, we need the second derivative to be zero. Let's find the second derivative of our curve:

(d^2/dx^2) ln(y) = (d/dx) (-2(x+5)/(b))
(1/y) (d^2y/dx^2) - (dy/dx)^2/y^2 = -2/b

Substituting x = -2 into the second derivative equation and solving for 'b':
(-2/b)*y = 0

Simplifying the equation, we get:
-2/b = 0

Again, this equation cannot hold true. Therefore, the curve of the given form does not have a point of inflection at x = -2.

To summarize, the formula for a curve with a local maximum at x = -5 and points of inflection at x = -8 and x = -2 does not exist in the form of y = e^((-((x-a)^2)/b)). The given conditions cannot be satisfied with this curve equation.