. An elementary course of infinitesimal calculus . ve is concave upwards when (x) cannot be either positiveor negative, and therefore (since it is assumed to be finite)must vanish. This condition, though essential, is not suffi-cient. It is further necessary that ^ («) should change sign 68] DEKIVATIVES OF HIGHEB ORDERS. 159 as X increases through the value in question. Suppose, forinstance, that to the left of P the curve lies below thetangent at P) and that to the right of P it lies above appears then from Art. 56 that there will be points of thecurve both to the right and to the left,
. An elementary course of infinitesimal calculus . ve is concave upwards when (x) cannot be either positiveor negative, and therefore (since it is assumed to be finite)must vanish. This condition, though essential, is not suffi-cient. It is further necessary that ^ («) should change sign 68] DEKIVATIVES OF HIGHEB ORDERS. 159 as X increases through the value in question. Suppose, forinstance, that to the left of P the curve lies below thetangent at P) and that to the right of P it lies above appears then from Art. 56 that there will be points of thecurve both to the right and to the left, in the immediateneighbourhood of P, at which the gradient is greater than atP, the gradient is a minimum at P, and <^ («) must,therefore change (Art. 50) from negative to Fig. 45. If the crossing is in the opposite direction, the gradientis a maximum at P, and f^ (x) changes from positive tonegative. Ex. 1. Ifwe have y = lox. ?(2), This changes from — to + as a; increases through 0. Hence wehave a point of inflexion ; see Fig. 34, p. 110. Ex. makes y= 2x y 1+3? „ _ 4a; (a; - 3)Q.+xf • .(3). 160 INFINITESIMAL CALCULUS. [CH. IV Hence there are three points of inflexion, viz. when x = 0 andwhen x = d= J3. See Fig. 17, p. 31. Ux. 3. In the curve of sines 2/ = &sin- (4), 1 II b . X 11 we have y — —; sin - = — -^. a a a Hence y changes sign, and there is a point of inflexion,whenever the curve crosses the axis of x. See Fig. 18, p. 34. Ex. 4. In the curve 2/ = a3* (5), we have y — \2i3?. This vanishes, but does not change sign, when a; = 0. Hencewe have a stationary tangent, but not a point of inflexion in thestrict sense. It is in fact obvious, since x* is essentiallypositive, that the curve lies wholly on one side of the tangenta
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