To state: The First derivative test.
The first derivative test:
“Suppose that c is a critical number of a continuous function f :
(a) If changes from positive to negative at c, then f has a local maximum at c.
(b) If changes from negative to positive at c, then f has a local minimum at c.
(c) If is positive to the left and right of c, or negative to the left and right of c, then f has no local maximum or minimum at c”.
To state: The second derivative test.
The second derivative test:
“Suppose. is continuous near c.
(a) If and, , then f has a local minimum at c.
(b) If and, then f has a local maximum at c”.
To explain: The relative advantages and disadvantages of first and second derivative tests.
First derivative test guarantees the local maxima or minima of a function at a point. But if the second derivative of a function is zero then this test fails to conclude about local maxima or minima.
First derivative test is time consuming and its process is lengthy. Also it is necessary to check the behavior of a function over the intervals. Whereas the second derivative test is less time consuming than the first derivative test.
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