critical points
f'(x) is either zero or undefined
Critical points of a function are the points at which the first derivative of the function is equal to zero or does not exist. In other words, these are the points where the slope of the function is either zero or undefined.
At a critical point, the function can have either a local minimum, a local maximum, or neither (a saddle point). To determine the nature of the critical point, we need to use the second derivative test, which involves evaluating the second derivative of the function at the critical point.
If the second derivative is positive at a critical point, the point corresponds to a local minimum. If the second derivative is negative, the point corresponds to a local maximum. If the second derivative is zero or undefined, the test is inconclusive, and we need to use additional methods to determine the nature of the critical point.
Identifying critical points is an important part of calculus, as it helps to locate the maximum and minimum values of a function, which are critical in many practical applications.
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