Let be a differentiable function and let be a critical point of .
1. If to the left of , and to the right of , then is a local minimum of .
2. If to the left of , and to the right of , then is a local maximum of .
3. In any other case, the critical point is neither a maximum or minimum. We call it a saddle point.
Example: Let . Find all critical points of and use the first derivative test to classify each point.
Solution: First, we find the derivative: . We set the derivative equal to zero to find the critical points:
So, the critical points are (−1,2), (0,0), and (1,−2). We apply the first derivative test by finding the derivatives of points to the left and the right of each critical point:
So what does this tell us? Well we look at each point individually.
For , the derivative changes from positive to negative. This tells us that is a local maximum.
For , the derivative is negative both to the left and to the right of the point. This tells us that is a saddle point.
For , the derivative changes from negative to positive. This tells us that is a local minimum.
We can examine the graph and confirm our findings:
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