calculus
Find the maximum and minimum values of a function
Find the maximum and minimum values of the function. The maximum and minimum values of a function can be found by analyzing its behavior.
Steps to find maximum and minimum values
1. Determine the domain of a function:
- Find where the function is defined. Check for points where the function might not exist (e.g., division by zero or square roots of negative numbers).
2. Find the critical points:
- Take the derivative of a function f′(x) and set it equal to zero.
Solve for x . These are the critical points.
- Also find where f′(x) is undefined within the domain, as these points can also indicate extrema.
3. Evaluate the endpoints (if applicable):
- If the function is specified on a closed interval [a,b], evaluate f(x) at the endpoints x=a and x=b.
4. Perform the second derivative test (optional):
Compute f′′(x) at the critical points:
- f ′′(x)>0 indicates that the point is a local minimum .
- f′′(x)<0 indicates that the point is a local maximum.
- If f′′(x)=0, the test is inconclusive, and other methods (like the first derivative test) are needed.
5. Compare values:
- Plug the critical points and end points (if applicable) into the original function f(x) to find their corresponding values .
- The minimum is the smallest value, and the maximum is the greatest.
How to find the maximum and minimum values of a function in a closed interval?
Example:
Find the maximum and minimum values of on [0, 3].
Step 1 : Domain
- The polynomial function f(x) is defined for all real numbers. There are no concerns with the interval [0, 3].
Step 2: Critical Points
- Consider the derivative: f ′(x) = 3x^2 −6x.
- Set f′(x)=0
3x^2−6x=0⇒3x(x−2)=0⇒x=0 or x=2.
critical points are x=0, x=2
Step 3: Endpoints
- Evaluate f(x) at x=0 and x=3
Step 4: Compare values
- ,
Step 5: Conclusion
- The maximum value is 4, appears at x = 0 and x = 3.
- The minimum value is 0, which appears at x = 2.