To determine whether a function has a minimum or maximum value, and to find that value, we generally follow these steps:

1. Differentiate the function:

Take the first derivative of the function to find the points where the slope of the curve is zero (critical points).

2. Set the derivative equal to zero:

Solve for $x$ where the derivative is zero. These are the potential minimum or maximum points.

3. Second Derivative Test:

• Take the second derivative of the function.
• Evaluate the second derivative at the critical points:
  • If $f''(x) > 0$, the point is a minimum.
  • If $f''(x) < 0$, the point is a maximum.
  • If $f''(x) = 0$, the test is inconclusive, and other methods like the First Derivative Test or analyzing the function's behavior around the critical points are needed.

Would you like to apply this process to a specific function or scenario? If you provide one, I can walk you through the steps.

Here are 5 example questions related to finding minimum or maximum values:

1. What is the maximum value of $f(x) = x^2 - 4x + 3$?
2. How do you find the minimum of $f(x) = -2x^2 + 8x - 5$?
3. Can a function have both a minimum and maximum value?
4. What is the significance of the second derivative in determining concavity?
5. How can you tell if a critical point is a saddle point?

Tip: When using the second derivative test, always check for concavity to ensure whether you are dealing with a maximum or a minimum.