Find the points of inflection.
f(x) = e^(-x^2)
I need to know step by step how you would get the points of inflection.
differentiate using the product rule, then differentiate again.
set that result equal to zero and solve for x
that x value is the x value of the point of inflection.
sub it back into the original to get the corresponding y value.
sorry,
there is no product rule here in the 1st derivative, but there will be one in the 2nd
I solved it for the second derivative. So I would use that to find the points?
Is the inflection point -2?
((e^(-x^2))'=-2xe^(-x^2)
((e^(-x^2))''=-2e^(-x^2)+4x^2e^(-x^2)=
2e^(-x^2)(2x^2-1)=0
Two points: (sqrt(1/2),e^(-1/2))
(-sqrt(1/2),e^(-1/2))
To find the points of inflection for the function f(x) = e^(-x^2), follow these step-by-step instructions:
Step 1: Find the second derivative of the function f(x).
In this case, the first derivative of f(x) is found by applying the chain rule:
f'(x) = d/dx [e^(-x^2)] = -2x * e^(-x^2)
Now, differentiate the first derivative to find the second derivative:
f''(x) = d/dx [-2x * e^(-x^2)] = -2e^(-x^2) + 4x^2 * e^(-x^2)
Step 2: Set the second derivative equal to zero.
To find the points of inflection, we need to determine the x-values where the second derivative is equal to zero:
-2e^(-x^2) + 4x^2 * e^(-x^2) = 0
Step 3: Factor out e^(-x^2).
Factor out e^(-x^2) from the above equation:
e^(-x^2) * (-2 + 4x^2) = 0
Step 4: Solve for x.
Set each factor equal to zero and solve for x:
-2 + 4x^2 = 0
4x^2 = 2
x^2 = 1/2
x = ±√(1/2)
Step 5: Determine the points of inflection.
The points of inflection on the graph of f(x) occur at the x-values obtained from the above step: x = ±√(1/2).
Therefore, the points of inflection for the function f(x) = e^(-x^2) are x = √(1/2) and x = -√(1/2).