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5.2 first and second derivative test

• 1. Relative Extrema First Derivative Test - FDT Second Derivative Test - SDT
• 3. Critical Points Critical points are points at which: •Derivative equals zero (also called stationary point). •Derivative doesn’t exist.
• 4. First Derivative Test Let f be a differentiable function with f '(c) = 0, then: •If f '(x) changes from positive to negative, then f has a relative maximum at c. •If f '(x) changes from negative to positive, then f has a relative minimum at c. •If f '(x) has the same sign from left to right, then f does not have a relative extremum at c.
• 5. Practice Time!!! Use First Derivative Test to find critical points and state whether they are 5 2 minimums or maximums. f ( x ) = 3x 3 −15x 3 2 −1 −1 2 1 − 5 3 2 3 3 3 f ' ( x ) = 3× x −15× x f ' ( x ) = 5x −10x = 5x 3 ( x − 2 ) 3 3 1 − 5 ( x − 2) 5x 3 ( x − 2 ) = 0 Critical points =0 1 x3 + 0 __ • 0 relative maximum • 2 relative minimum + 2(stationary)
• 6. Second Derivative Test Suppose that c is a critical point at which f’(c) = 0, that f(x) exists in a neighborhood of c, and that f(c) exists. Then: • f has a relative maximum value at c if f”(c) < 0. •f has a relative minimum value at c if f”(c) > 0. •If f(c) = 0, the test is not conclusive. Note: Second derivative test is still used to calculate max and min
• 7. Practice Time again !!! Use second derivative test to find extrema of f (x) = 3x 5 − 5x 3 f '(x) = 15x − 15x 4 2 15x − 15x = 0 4 2 critical points = 0, -1, 1 f "(x) = 60x 3 − 30x inconclusive f "(0) = 0 f "(− 1) = − 30 < 0 f has a maximum at x = -1 f has a minimum at x = 1 f "(1) = 30 > 0 15x 2 ( x 2 − 1) = 0
• 8. x2 Find extrema of f (x) = x 4 +16 A.Using first derivative test B.Using second derivative test
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