Final answer:
To solve for a relative minimum, you can use the first derivative test, the second derivative test, or critical point analysis. These methods involve finding critical points, evaluating derivatives, and comparing signs to determine if the critical points are relative minimums or maximums.
Step-by-step explanation:
Solving for Relative Minimum
To solve for a relative minimum, we can use the first derivative test, the second derivative test, or critical point analysis. Let's go through each method:
A: First Derivative Test
Find the first derivative of the function.
Determine the critical points by setting the first derivative equal to zero or undefined.
Evaluate the first derivative at points on each side of the critical points.
Check the signs of the first derivative to determine if the critical points are relative minimums or maximums.
B: Second Derivative Test
Find the second derivative of the function.
Determine the critical points by setting the second derivative equal to zero or undefined.
Evaluate the second derivative at points on each side of the critical points.
Check the signs of the second derivative to determine if the critical points are relative minimums or maximums.
C: Critical Point Analysis
Find the critical points by setting the derivative equal to zero or undefined.
Evaluate the function at each critical point.
Compare the function values to determine if the critical points are relative minimums or maximums.
Example:
Let's say we have the function f(x) = 2x^3 - 12x^2 + 18x. Using the first derivative test, we find that the critical points are x = 0 and x = 3. Evaluating the first derivative at points near these critical points, we find that f'(x) is negative to the left of x = 0 and positive to the right of x = 3, indicating that x = 0 is a relative maximum and x = 3 is a relative minimum.