Second Order Conditions

We saw in section 6.! that the condition f'(x") = 0 does not in itself tell us whether x* yields a maximum, a minimum, or a point of inflection of the function /. Since this condition is staled in terms of the first derivative of the function it is usually referred to as the first-order condition. We now go on to examine how conditions on the second derivative of a function, namely second-order conditions, can be developed to help us distinguish among the three kinds of stationary value.

In figure 6.16 we show the three possible cases of stationary values. Associated with each graph of a function / is. directly below it. the graph of its first

(a)
/'(.**) = 0
(e>

fix)

rU)

f'Xx*} > 0

AO'

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