A critical number is an input value (x-value) where the derivative of a function is either zero or undefined. There isn't a "critical value of a critical number" in the standard mathematical sense. The value of the function at a critical number, however, can be highly significant, as it may represent a local maximum, a local minimum, or a saddle point.
Here's a breakdown to clarify:
-
Critical Number (c): This is an x-value (in the domain of the function) where f'(c) = 0 or f'(c) is undefined.
-
Critical Point: This is the point (c, f(c)), where 'c' is a critical number and f(c) is the function's value at that point.
-
Critical Value (f(c)): This is the y-value of the function at a critical number. It is the function's value, f(c), evaluated at the critical number 'c'. This value could be a local maximum, local minimum, or neither.
To find critical values:
- Find the derivative: Calculate f'(x) of the given function f(x).
- Find critical numbers: Determine the values of x where f'(x) = 0 or f'(x) is undefined.
- Evaluate the function at the critical numbers: Calculate f(c) for each critical number 'c'. The result, f(c), is the critical value.
Example:
Let's say we have the function f(x) = x3 - 3x.
- Find the derivative: f'(x) = 3x2 - 3
- Find critical numbers:
- Set f'(x) = 0: 3x2 - 3 = 0 => x2 = 1 => x = 1 or x = -1.
- f'(x) is defined for all x, so there are no other critical numbers.
- Evaluate the function at the critical numbers:
- f(1) = (1)3 - 3(1) = -2
- f(-1) = (-1)3 - 3(-1) = 2
Therefore, the critical numbers are 1 and -1, and the critical values are -2 and 2. These values represent the y-values of the critical points (1, -2) and (-1, 2).
In short, the critical value is the function's output (y-value) obtained when a critical number (x-value) is plugged into the original function.
