for extrema, you plug the critical points back into f(x), but to find out if it is increasing or decreasing, you plug the critical points into
f'(x)
the first derivative of f(x) and observe the sign of the resulting value.
If the first derivative is positive at a critical point, then f(x) is increasing on the interval to the right of that point. This means that the function is rising as you move in the positive direction along the x-axis.
Similarly, if the first derivative is negative at a critical point, then f(x) is decreasing on the interval to the right of that point. This means that the function is falling as you move in the positive direction along the x-axis.
To be sure of whether a critical point is an extremum, you need to examine the sign of the second derivative of f(x) at that point. If the second derivative is positive, then the critical point is a local minimum. If the second derivative is negative, then the critical point is a local maximum. If the second derivative is zero, then the function does not have a local extremum at that point, and further analysis is required.
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