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.

Got it....

Why did the curve go see a psychiatrist? Because it had too many inflection points and couldn't function properly!

Now, let's get down to business. To satisfy the given conditions, we can start by using the general form of the curve, y = e^(-(x - a)^2 / b).

For a local maximum at x = -5, we can set x = -5 and find that the first derivative equals zero:

dy/dx = (-2(x - a) / b) * e^(-(x - a)^2 / b) = 0

Simplifying this gives -2(x - a) = 0, which implies x = a.

Since we want a maximum at x = -5, we set a = -5.

Next, for points of inflection at x = -8 and x = -2, we can set x = -8 and x = -2 in the second derivative and find that it equals zero:

d²y/dx² = ((2(x - a)^2)/b^2 - 1/b) * e^(-(x - a)^2 / b) = 0

Using the value a = -5, we can plug in x = -8 and x = -2 to obtain two equations:

((2(-8 + 5)^2)/b^2 - 1/b) * e^(-(-8 + 5)^2 / b) = 0

((2(-2 + 5)^2)/b^2 - 1/b) * e^(-(-2 + 5)^2 / b) = 0

By solving these equations simultaneously, we can find the value of b.

Now, if only we could find a formula to solve all of life's inflection points...

To find a formula for the curve of the form y = e^((-1(x-a)^2)/b), we need to determine the values of a and b based on the given information.

1. Local Maximum:
The curve has a local maximum at x = -5. At this point, the derivative of the curve with respect to x is equal to 0.

First, let's find the derivative(dy/dx) of the given curve:

dy/dx = d/dx [e^((-1(x-a)^2)/b)]

Using the chain rule, we have:

dy/dx = (-2(x-a)/b) * e^((-1(x-a)^2)/b)

Setting dy/dx equal to zero and solving for x:

0 = (-2(x-a)/b) * e^((-1(x-a)^2)/b)

Since e^((-1(x-a)^2)/b) is always positive for any value of x, we can ignore it and solve for x-a:

0 = -2(x-a)/b

0 = -2(x-a)

2(x-a) = 0

x - a = 0

x = a

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

2. Points of Inflection:
The curve has points of inflection at x = -8 and x = -2. At these points, the second derivative of the curve with respect to x changes sign.

Let's find the second derivative (d²y/dx²) of the given curve:

d²y/dx² = d/dx [-2(x-a)/b * e^((-1(x-a)^2)/b)]

d²y/dx² = [2(x-a)² - b]/b² * e^((-1(x-a)^2)/b)

Setting d²y/dx² equal to zero and solving for x:

0 = [2(x-a)² - b]/b² * e^((-1(x-a)^2)/b)

Since e^((-1(x-a)^2)/b) is always positive for any value of x, we can ignore it and solve for x-a:

0 = 2(x-a)² - b

2(x-a)² = b

(x-a)² = b/2

Taking the square root of both sides:

x - a = sqrt(b/2) or x - a = -sqrt(b/2)

Therefore, the points of inflection occur at x = a + sqrt(b/2) = -8 and x = a - sqrt(b/2) = -2.

From all the information we found:

- The local maximum occurs at x = a = -5.
- The points of inflection occur at x = a + sqrt(b/2) = -8 and x = a - sqrt(b/2) = -2.

Now, substituting the x-values for a, let's find the value of b:

-8 = -5 + sqrt(b/2)

-8 + 5 = sqrt(b/2)

-3 = sqrt(b/2)

Square both sides:

9 = b/2

b = 18

Therefore, the formula for the curve is:

y = e^((-1(x-(-5))^2)/18)

Simplifying further:

y = e^((-1(x+5)^2)/18)

To find a formula for the curve of the given form, we need to determine the values of the parameters a and b. Since the given curve has a local maximum at x = -5 and points of inflection at x = -8 and x = -2, we can use this information to find the values of a and b.

Let's start by looking at the local maximum. For any curve of the form y = e^((-1/b)(x-a)^2), the x-coordinate of the local maximum is equal to a. In this case, we are given that the local maximum occurs at x = -5. Thus, we have a = -5.

Next, let's analyze the points of inflection. Points of inflection occur when the second derivative of the function changes sign. The second derivative of y = e^((-1/b)(x-a)^2) with respect to x can be calculated as follows:

d²y/dx² = (2/b^2)(x-a)^2 * e^((-1/b)(x-a)^2)

To find the points of inflection, we set the second derivative equal to zero and solve for x:

0 = (2/b^2)(x-a)^2 * e^((-1/b)(x-a)^2)

Since e^((-1/b)(x-a)^2) is always positive, the only way for this equation to be true is if (x-a)^2 = 0. Therefore, x = a.

We are given that the points of inflection occur at x = -8 and x = -2. Thus, we have two equations:

-8 = -5 (equation 1)
-2 = -5 (equation 2)

From equation 1, we find that b = (-8 - (-5))^2 = 9.
From equation 2, we find that b = (-2 - (-5))^2 = 9.

Since both equations give us the same value for b, we can conclude that b = 9.

Therefore, the formula for the curve of the given form is:

y = e^((-1/9)(x-(-5))^2)

Simplifying further, we have:

y = e^((-1/9)(x+5)^2)

This equation represents a curve of the given form with a local maximum at x = -5 and points of inflection at x = -8 and x = -2.