Analyzing Critical Points, Endpoints, and Function Behavior: Finding Maximum and Minimum Values of a Function

Find max/min value

To find the maximum and minimum values of a function, we need to analyze its critical points, endpoints, and the behavior of the function

To find the maximum and minimum values of a function, we need to analyze its critical points, endpoints, and the behavior of the function.

1. Critical Points: These occur when the derivative of the function is equal to zero or undefined. To find critical points, follow these steps:
a) Take the derivative of the function.
b) Set the derivative equal to zero and solve for x.
c) If the derivative is undefined at any point, consider those points as well.

2. Endpoints: If the function is defined over a closed interval, check the values of the function at the endpoints.

3. Behavior:
a) For a continuous function defined on a closed interval, evaluate the function at the critical points and endpoints.
b) Compare the values obtained to determine the maximum and minimum values.

Let’s work through an example to illustrate these steps:

Example: Find the maximum and minimum values for the function f(x) = 2x^3 – 6x^2 + 4x on the interval [0, 3].

Solution:
1. Critical Points:
a) Take the derivative of f(x): f'(x) = 6x^2 – 12x + 4.
b) Set the derivative equal to zero and solve for x:
6x^2 – 12x + 4 = 0
Applying the quadratic formula or factoring, we get:
x = (3 ± √5)/3
So, the critical points are x = (3 – √5)/3 and x = (3 + √5)/3.

2. Endpoints: Since the interval is [0, 3], the endpoints are 0 and 3.

3. Behavior:
a) Evaluate the function at the critical points and endpoints:
f(0) = 2(0)^3 – 6(0)^2 + 4(0) = 0
f((3 – √5)/3) = 2((3 – √5)/3)^3 – 6((3 – √5)/3)^2 + 4((3 – √5)/3)
f((3 + √5)/3) = 2((3 + √5)/3)^3 – 6((3 + √5)/3)^2 + 4((3 + √5)/3)
f(3) = 2(3)^3 – 6(3)^2 + 4(3)
b) Compare the function values to determine the maximum and minimum:
From the values obtained, the maximum and minimum values of the function on the given interval can be identified.

Remember, this is just the general procedure for finding maximum and minimum values. Different functions may require different approaches, and sometimes additional techniques like the second derivative test may be needed.

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