. Applied calculus; principles and applications . The slope f (x) is positive immediately before a maximumordinate, and negative immediately after; while the slope isnegative immediately before a minimum ordinate and posi-tive after. The slope / (x) is 0 or 00 at any point whoseordinate / (x) is either a maximum or a minimum. Theslope / (x) is discontinuous at the points P4 and P5, whereit changes sign by becoming infinite as x increases throughthe values OM4 and OM5; that is, / (a;) = 114 DIFFERENTIAL CALCULUS The slope f {x) is 0 at Pe and oo at P7; but it does notchange sign at either
. Applied calculus; principles and applications . The slope f (x) is positive immediately before a maximumordinate, and negative immediately after; while the slope isnegative immediately before a minimum ordinate and posi-tive after. The slope / (x) is 0 or 00 at any point whoseordinate / (x) is either a maximum or a minimum. Theslope / (x) is discontinuous at the points P4 and P5, whereit changes sign by becoming infinite as x increases throughthe values OM4 and OM5; that is, / (a;) = 114 DIFFERENTIAL CALCULUS The slope f {x) is 0 at Pe and oo at P7; but it does notchange sign at either point, and neither M&P^ nor M^P^ is amaximum or a minimum ordinate; it does, however, changein value at each point, Pe being a point where the slope f (x) isa minimum and P7 one where it is a maximum. The pointsPe and P? are inflexion points, at which the curve changesfrom being concave downward to upward, or vice versa. Note. — Points such as P4 and P5 occur on railroad ^ Ys,and such points where branches of a curve end tangent to each other are called a point on a non-algebraiccurve where branches end andare not tangent to each other,called a shooting point, f (x)may change abruptly from apositive finite value to a nega-tive value, or vice versa; hence,/ (a) would be a maximum or a minimum without / (x)becoming either zero or infinite. The supplementary figureshows a shooting point at which / (a) is a minimum; / (x)becoming — 1 as x increases to a, and +1 as x dec
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