What are the points of inflection, if any, of #f(x) = x^4/12 - 2x^2 + 15 #?
2 Answers
The function
Explanation:
A necessary condition for the curve
Evaluate the second derivative of the function:
From the equation:
we get that the function may have an inflection point in
Now consider the inequality:
As
There are two non-stationary points of inflection which occur at
Explanation:
We have:
# f(x) = x^4/12-2x^2+15 #
We would normally look for critical points, that is coordinates where
The first derivative is then:
# f'(x) = x^3/3-4x #
So the second derivative is then:
# f''(x) = x^2-4 #
We look for inflection points , which are coordinates where the second derivative vanishes:
# f''(x) = 0 => x^2-4 = 0 #
# :. x^2=4 #
# :. x = +-2 #
A point of inflection is graded as a stationary point of inflection of the first derivative vanished at the point, otherwise as a non-stationary point of inflection
So when
# f(-2) = 16/12-8+15 = 25/3 #
# f'(-2) = -8/3+8 = 16/3 #
And when
# f'(2) = 16/12-8+15=25/3#
# f'(2) = 8/3-8 = 16/3#
Hence, There are two non-stationary points of inflection which occur at
It can be interesting to see the graphs of the function compared with the first and second derivatives:
The graph of the function
graph{x^4/12-2x^2+15 [-6, 6, -10, 18]}
The graph of the function
graph{x^3/3-4x [-6, 6, -10, 18]}
The graph of the function
graph{x^2-4 [-6, 6, -10, 18]}