## Criteria For Relative Extrema

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# Criteria For Relative Extrema

Let f ‘(x

1. If n is odd, f has neither a relative maximum nor a relative minimum at x = x

2. If n is even, f has a relative maximum if f

This method is illustrated in the following examples.

f ‘ (x) = 6x

Now f ‘ (x) = 0 when x = 2 and x = 3

Also f “ (x) = 12x – 30

f “ (2) = 12 (2) – 30 = - 6 < 0

hence there is a relative maximum at x = 2.

Further f “ (3) = 12 (3) – 30 = 6 > 0,

hence there is a relative minimum at x = 3.

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_{0}) = f ‘’ (x_{0}) = ... = f^{(n-1)}(x_{0}) = 0, but f^{(n)}(x_{0}) ≠ 0. Then1. If n is odd, f has neither a relative maximum nor a relative minimum at x = x

_{0}.2. If n is even, f has a relative maximum if f

^{(n)}(x_{0}) < 0, and has a relative minimum if f^{ (n)}(x_{0}) > 0.This method is illustrated in the following examples.

**Example.**Use the second derivative test to find the relative maxima and minima of f (x) = 2x^{3 }– 15x^{2}+ 36x + 18.**Solution.**Differentiating the given function with respect to x, we obtainf ‘ (x) = 6x

^{2}– 30x + 36 = 6 (x-2) (x-3)Now f ‘ (x) = 0 when x = 2 and x = 3

Also f “ (x) = 12x – 30

f “ (2) = 12 (2) – 30 = - 6 < 0

hence there is a relative maximum at x = 2.

Further f “ (3) = 12 (3) – 30 = 6 > 0,

hence there is a relative minimum at x = 3.

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