Find the absolute minimum of f on -10,7
Answer
Finding the Absolute Minimum of a Function on a Closed Interval
To determine the absolute minimum of a continuous function on a closed interval, we follow a structured process using calculus and comparison of values at critical points and endpoints.
Step 1: Compute the Derivative of f(x)
The derivative, f'(x), helps locate where the function might reach a local extremum (minimum or maximum). These are called critical points.
Step 2: Find Critical Points Inside the Interval
Solve f'(x) = 0 and check for points where f'(x) is undefined within the interval [−10, 7]. Only those values inside the interval are relevant.
Step 3: Evaluate f(x) at Critical Points and Endpoints
To find the absolute minimum, calculate the function values at:
- Each critical point within the interval
- The endpoints of the interval:
x = −10andx = 7
Compare all results.
Step 4: Identify the Absolute Minimum Value
The smallest value among those evaluated in Step 3 is the absolute minimum.
Note: The actual result depends on the expression of f(x), which you’ll need to plug in and evaluate.