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 the curve with the given properties, we need to determine the values of a and b in the equation y = e^((-(x-a)^2)/b).

Step 1: Local maximum at x = -5
At the local maximum, the derivative of the curve is zero. Let's find the derivative and set it equal to 0.
dy/dx = -2(x - a) / b * e^((-(x - a)^2) / b) = 0

Canceling the common factor, we get:
-2(x - a) = 0

Solving for x, we have:
x - a = 0
x = a

So, the local maximum occurs at x = a = -5.

Step 2: Points of inflection at x = -8 and x = -2
At the points of inflection, the second derivative of the curve is zero. We need to find the second derivative and set it equal to 0.
d²y/dx² = (2(x - a)^2 - b) / b^2 * e^((-(x - a)^2) / b) = 0

Multiply both sides by b^2 and simplify:
2(x - a)^2 - b = 0

Expanding the equation, we get:
2x^2 - 4ax + (2a^2 - b) = 0

Comparing the coefficients with a quadratic equation ax^2 + bx + c = 0, we have:
a = 2
b = 2a^2 = 8

Step 3: Final formula
Therefore, the formula for the curve is:
y = e^((-(x - (-5))^2) / 8)

Simplifying further:
y = e^((-(x + 5)^2) / 8)

This formula represents a curve with a local maximum at x = -5 and points of inflection at x = -8 and x = -2.

To find a formula for the curve of the form y = e^((-(x - a)^2)/b), we can use the given conditions.

1. Local maximum at x = -5:
For a local maximum at x = -5, the derivative of the function should be zero at x = -5. Let's differentiate the function with respect to x:

dy/dx = (2(x - a)/b)e^((-(x - a)^2)/b)

Setting x = -5 and equating dy/dx to zero, we have:

0 = (2(-5 - a)/b)e^((-( -5 - a)^2)/b)

Simplifying the equation, we get:

2(-5 - a)e^((-( -5 - a)^2)/b) = 0

Since b is greater than 0, e^((-( -5 - a)^2)/b) will never be zero. Therefore, for the equation to hold true, we must have:

2(-5 - a) = 0

Solving for a, we find:

a = -5

2. Points of inflection at x = -8 and x = -2:
For points of inflection, the second derivative of the function should change sign at x = -8 and x = -2. Let's find the second derivative of the function:

d^2y/dx^2 = ((2(b - 2(x - a)^2))/(b^2))e^((-(x - a)^2)/b)

For x = -8, we have:

d^2y/dx^2 = ((2(b - 2(-8 - a)^2))/(b^2))e^((-( -8 - a)^2)/b)

For x = -2, we have:

d^2y/dx^2 = ((2(b - 2(-2 - a)^2))/(b^2))e^((-( -2 - a)^2)/b)

For the function to have points of inflection at these x-values, we need the second derivative to change sign at these points:

((2(b - 2(-8 - a)^2))/(b^2))e^((-( -8 - a)^2)/b) = -((2(b - 2(-2 - a)^2))/(b^2))e^((-( -2 - a)^2)/b)

Simplifying the equation, we get:

(b - 2(-8 - a)^2)e^((-( -8 - a)^2)/b) = -(b - 2(-2 - a)^2)e^((-( -2 - a)^2)/b)

Since b is greater than 0, e^((-( -8 - a)^2)/b) and e^((-( -2 - a)^2)/b) will never be zero. Therefore, for the equation to hold true, we must have:

b - 2(-8 - a)^2 = -(b - 2(-2 - a)^2)

Expanding and simplifying further, we find:

b + 128 + 32a + 2a^2 = b + 4 + 4a + 2a^2

Simplifying the equation, we get:

128 + 32a = 4 + 4a

Subtracting 4a from both sides and simplifying again, we find:

32a = -124

Dividing by 32, we find:

a = -124/32

Simplifying, we get:

a = -31/8

Therefore, the formula for the curve with a local maximum at x = -5 and points of inflection at x = -8 and x = -2 is:

y = e^((-(x - (-31/8))^2)/b)