Find the inflection points of f(x)=(x^2-6x-5)e^x
To find the inflection points of a function, you need to follow these steps:
Step 1: Find the second derivative of the function.
Step 2: Set the second derivative equal to zero and solve for x.
Step 3: The values of x obtained from Step 2 are the potential inflection points of the function.
Step 4: To determine if these potential inflection points are actual inflection points, you need to check the concavity of the function around each point.
Let's apply these steps to the function f(x) = (x^2-6x-5)e^x.
Step 1: Find the second derivative
To find the second derivative, we start by finding the first derivative of f(x):
f'(x) = (2x - 6)e^x + (x^2 - 6x - 5)e^x
Now, let's find the second derivative by differentiating f'(x):
f''(x) = (2x - 6)e^x + (2x - 6)e^x + (x^2 - 6x - 5)e^x
Simplifying, we get:
f''(x) = (4x - 12)e^x + (x^2 - 6x - 5)e^x
Step 2: Set the second derivative equal to zero and solve for x
To find the potential inflection points, we set f''(x) equal to zero:
(4x - 12)e^x + (x^2 - 6x - 5)e^x = 0
Factoring out e^x:
e^x (4x - 12 + x^2 - 6x - 5) = 0
Simplifying:
e^x (x^2 - 2x - 17) = 0
To solve this equation, we set each factor equal to zero:
e^x = 0 or x^2 - 2x - 17 = 0
The equation e^x = 0 has no real solutions because e^x is always positive.
To solve x^2 - 2x - 17 = 0, we can either factor or use the quadratic formula. In this case, factoring is not possible, so let's use the quadratic formula.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 1, b = -2, and c = -17, we have:
x = (-(-2) ± √((-2)^2 - 4(1)(-17))) / (2(1))
x = (2 ± √(4 + 68)) / 2
x = (2 ± √72) / 2
x = (2 ± 2√18) / 2
x = 1 ± √18
Step 3: Determine the potential inflection points
The potential inflection points are:
x = 1 + √18
x = 1 - √18
Step 4: Check the concavity around each potential inflection point
To determine if these potential inflection points are actual inflection points, we need to check the concavity of the function f(x) around each point.
To check concavity, we can evaluate the second derivative f''(x) at each potential inflection point.
For x = 1 + √18:
f''(1 + √18) = (4(1 + √18) - 12)e^(1 + √18) + ((1 + √18)^2 - 6(1 + √18) - 5)e^(1 + √18)
For x = 1 - √18:
f''(1 - √18) = (4(1 - √18) - 12)e^(1 - √18) + ((1 - √18)^2 - 6(1 - √18) - 5)e^(1 - √18)
By evaluating the second derivative at these points, we can determine if the function changes concavity, and therefore, if they are inflection points.