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 given curve, we need to determine the values of 'a' and 'b' in the equation y = e^(-((x-a)^2)/b).
Let's start by analyzing the given information:
1. Local maximum at x = -5: A local maximum occurs when the derivative of the function equals zero. So, we need to find the derivative of the curve.
dy/dx = (2 * (x - a) * e^(-((x - a)^2)/b)) / b
To have a local maximum at x = -5, we substitute x = -5 into the derivative and set it equal to zero:
(2 * (-5 - a) * e^(-((-5 - a)^2)/b)) / b = 0
This implies that the term (-5 - a) must be equal to zero:
-5 - a = 0
a = -5
2. Points of inflection at x = -8 and x = -2: Points of inflection occur when the second derivative of the function equals zero or changes sign. We need to find the second derivative.
Taking the derivative of the already derived derivative:
d^2y/dx^2 = (2 * e^(-((x - a)^2)/b)) / (b^2) * (b - 2 * (x - a)^2)
Now, let's determine the points of inflection:
For x = -8:
(2 * e^(-((-8 - a)^2)/b)) / (b^2) * (b - 2 * (-8 - a)^2) = 0
For x = -2:
(2 * e^(-((-2 - a)^2)/b)) / (b^2) * (b - 2 * (-2 - a)^2) = 0
However, trying to solve these equations analytically may be quite challenging and may not lead to a simple formula for 'b'. Numerical methods or approximation techniques might be required to find a suitable value for 'b'.
In summary, we have found that 'a' is equal to -5, but finding an exact formula for 'b' may require using numerical methods or approximation techniques.