Critical numbers
First derivative equals zero or does not exist
Critical numbers are an important concept in calculus that is used to find the maximum or minimum points of a function. A critical number is a point at which either the derivative of the function is zero or it does not exist.
To find the critical numbers of a function, we first differentiate the function and solve the resulting equation to find the points where the derivative is zero or undefined. These points can then be analyzed to determine whether they correspond to a maximum or minimum point of the function.
For example, consider the function f(x) = x^3 – 3x^2 + 2. To find the critical numbers, we first differentiate the function with respect to x:
f'(x) = 3x^2 – 6x
We then set this derivative equal to zero and solve for x:
3x^2 – 6x = 0
3x(x – 2) = 0
This equation has two solutions: x = 0 and x = 2. These are the critical numbers of the function.
To determine whether these points correspond to a maximum or minimum point of the function, we can use the second derivative test. We take the second derivative of the function:
f”(x) = 6x – 6
And evaluate it at each critical number.
f”(0) = -6 < 0, so x = 0 corresponds to a local maximum. f''(2) = 6 > 0, so x = 2 corresponds to a local minimum.
Therefore, the critical numbers of the function f(x) are x = 0 and x = 2, and they correspond to a local maximum and local minimum, respectively.
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