relative min/max
the lowest/highest point of a graph within a prescribed interval on the graph
Relative minimum and maximum refer to turning points on a graph which occur on a part of the graph where the slope of the graph changes sign from upward to downward or vice versa. A relative minimum is a turning point on a graph where the function changes from decreasing to increasing, and a relative maximum is a turning point on a graph where the function changes from increasing to decreasing.
To find relative maxima and minima, we need to follow these steps:
1. Determine the first derivative of the function.
2. Determine the critical points of the function by finding where the first derivative is equal to zero or undefined.
3. Determine the sign of the first derivative in the intervals between the critical points.
4. Analyze the sign of the first derivative in each interval and determine whether the function is increasing or decreasing.
5. Check the behavior of the function at the endpoints of the interval. If necessary, evaluate the function at the critical points, and compare the values obtained with those at the endpoints.
6. Identify relative maxima and minima based on the intervals where the function changes from increasing to decreasing or vice versa.
For example, let’s say we have the function f(x) = x^3 – 3x^2 – 9x + 10, and we want to find its relative maxima and minima.
1. Determine the first derivative of the function.
f'(x) = 3x^2 – 6x – 9
2. Determine the critical points of the function by finding where the first derivative is equal to zero or undefined.
f'(x) = 0 when x = -1 and x = 3.
3. Determine the sign of the first derivative in the intervals between the critical points.
Interval (-∞, -1):
f'(-2) = 15 > 0, so f(x) is increasing.
Interval (-1, 3):
f'(0) = -9 < 0, so f(x) is decreasing.
Interval (3, ∞):
f'(4) = 39 > 0, so f(x) is increasing.
4. Analyze the sign of the first derivative in each interval and determine whether the function is increasing or decreasing.
The function is increasing from negative infinity to -1, decreasing from -1 to 3, and increasing again from 3 to infinity.
5. Check the behavior of the function at the endpoints of the interval. If necessary, evaluate the function at the critical points, and compare the values obtained with those at the endpoints.
The function tends to negative infinity as x tends to negative infinity, and tends to infinity as x tends to infinity.
f(-1) = 13 and f(3) = -7.
6. Identify relative maxima and minima based on the intervals where the function changes from increasing to decreasing or vice versa.
We have a relative maximum at x = -1 and a relative minimum at x = 3.
Therefore, the relative maximum of the function f(x) = x^3 – 3x^2 – 9x + 10 is (-1, 13) and the relative minimum is (3, -7).
More Answers:
Understanding the Characteristics of Exponential Functions: f(x) = a^x for Positive Constant ‘a’Partial Differentiation: A Guide to Calculating Partial Derivatives of Functions with Respect to Multiple Variables.
The Complete Guide to Using the Chain Rule in Calculus for Multiple Composed Functions