Stationary Points Of A Function (local max/min etc.) (Calculus)
By DarthVader
Date: 2022-08-12
Topic: 156 see comments
Post views: 1036
Stationary points of a function (no slope)
A stationary point could be a local minimum/maximum or a point of inflection.
Finding the stationary points of a function
To find the stationary points of a function first take the derivative and then solve the equation f '(x) = 0,
For example, if:
f(x) = 3x2 + 4x − 6
Then the derivative is:
f '(x) = 6x + 4
So to find the stationary points, just set 6x + 4 equal to zero:
6x + 4 = 0
This can be done using the quadratic formula, and the solutions for ‘x’ will be the ‘stationary points’ of the function.
x = −4/6
So this function has one stationary (critical) point and that is: −4/6
First derivative test (for determining the nature of a stationary point of a function f)
If a function continues on both the left side and the right side of a stationary point, then:
- if f '(x) is positive on the left side and negative on the right side, the stationary point is a local maximum
- if f '(x) is negative on the left side and positive on the right side, the stationary point is a local minimum
- if f ‘(x) is positive on both sides or negative on both sides, the stationary point is a horizontal point of inflection (think graph of trig function ’tan')
Applying the first derivative test by choosing sample points
1. Choose two points (that is, two x-values) fairly close to the stationary point, one on each side.
2. Check that the function is differentiable at all points between the chosen points and the stationary point, and that there are no other stationary points between the chosen points and the stationary point.
3. Find the derivative of the function at the two chosen points.
- If the derivative is positive at the left chosen point and negative at the right chosen point, the stationary point is a local maximum.
- If the derivative is negative at the left chosen point and positive at the right chosen point, the stationary point is a local minimum.
- If the derivative is positive at both chosen points or negative at both chosen points, the stationary point is a horizontal point of inflection.
Example:
If you have a function like:
f(x) = (4/3)x3 + 5x2 − 6x − 6
Then the derivative is:
f '(x) = 4x2 + 10x − 6
So using the quadratic formula, the stationary points are:
[x = 0.5, x = −3]
So choose two points either side of ‘0.5’ such as, −1 and, 1
Then take the derivative of these two points by plugging them into f '(x) = 4x2 + 10x − 6:
f '(−1) = 4(−1)2 + 10(−1) − 6 = −12
f '(1) = 4(1)2 + 10(1) − 6 = 8
So since f ‘(−1) which is the point on the left side of the point ’0.5', is negative, and f ‘(1) which is the point on the right side of the point ’0.5', is positive, this means the curve travels down and then up and so the stationary point ‘0.5’ is a local minimum, where the curve dips.
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