How do i find the points of inflection for: f(x)= (1/12)x^4 -2x^2 +15
The answers are (2, 25/3) and (-2, 25/3) but i don't know how to get the answers. Can someone show me step by step?
You will need the second derivative, which is easily found to be x^2 - 4
setting this equal to zero gives us
x^2 = 4
x = ±2
plug x = 2 back into the original to get f(2) = (1/12)(16) - 2(4) + 15
= 25/3
do the same for x = -2
and you are done.
To find the points of inflection for the function f(x) = (1/12)x^4 - 2x^2 + 15, you need to follow these steps:
Step 1: Find the second derivative of the function.
To determine the points of inflection, you need to find the second derivative of the function. Start by finding the first derivative and then differentiate again.
Given f(x) = (1/12)x^4 - 2x^2 + 15,
First derivative: f'(x) = (4/12)x^3 - 4x
Second derivative: f''(x) = (12/12)x^2 - 4
Simplifying the second derivative gives: f''(x) = x^2 - 4
Step 2: Set the second derivative equal to zero and solve for x.
For points of inflection, the second derivative must equal zero. Therefore, you need to solve the equation x^2 - 4 = 0.
To solve x^2 - 4 = 0, add 4 to both sides: x^2 = 4
Taking the square root of both sides gives: x = ±2
So, the potential points of inflection are x = 2 and x = -2.
Step 3: Find the corresponding y-values for the potential points of inflection.
To find the corresponding y-values for the potential points of inflection, substitute the x-values into the original function f(x).
For x = 2, plug it into f(x): f(2) = (1/12)(2^4) - 2(2^2) + 15
Simplifying gives: f(2) = (1/12)(16) - 2(4) + 15 = (1/12)(16) - 8 + 15 = 16/12 - 8 + 15 = 4/3 + 7 = 25/3
Therefore, one of the points of inflection is (2, 25/3).
Now, for x = -2, plug it into f(x): f(-2) = (1/12)(-2^4) - 2(-2^2) + 15
Simplifying gives: f(-2) = (1/12)(16) - 2(4) + 15 = (1/12)(16) - 8 + 15 = 16/12 - 8 + 15 = 4/3 + 7 = 25/3
Therefore, the second point of inflection is (-2, 25/3).
So, the points of inflection for the function f(x) = (1/12)x^4 - 2x^2 + 15 are (2, 25/3) and (-2, 25/3).