. Differential and integral calculus. inimum , at a minimum .value, fix) = o, or f(x) = appears, therefore, that the essential condition for a maximumor minimum value of a function of a single variable is that itsfirst derivative shall change sign, — in case of a maximum value,from + to —; of a minimum value, from — to +. It also ap-pears that the value, or values, of x which render fix) a maxi-mum or minimum will be found among the roots of the equa-tions formed by equating the first derivative to zero or to infin-ity. These roots are called critical values of the variable, a
. Differential and integral calculus. inimum , at a minimum .value, fix) = o, or f(x) = appears, therefore, that the essential condition for a maximumor minimum value of a function of a single variable is that itsfirst derivative shall change sign, — in case of a maximum value,from + to —; of a minimum value, from — to +. It also ap-pears that the value, or values, of x which render fix) a maxi-mum or minimum will be found among the roots of the equa-tions formed by equating the first derivative to zero or to infin-ity. These roots are called critical values of the variable, andmust be separately examined, in order to ascertain which, ifany, give rise to a maximum or minimum state of the function. 113. Illustration. — Since fix), when considered geometri-cally, always represents the slope of the tangent to the curvey =f(x) (§ 19), the principles of the preceding article may begraphically represented. I. Critical values which renderf(x) = o. Let SM (Fig. 15) be the locus of the equation y=f(x). -M. At the maximum and minimum points of the curve, D and E, andat such points as F where the direction of curvature changes, thetangents are II to the #-axis; Maxima and Minima 141 henC6 dl=f(x) = o ax x for the critical values OA, OB, and OC, of x. For a value of x a little less than OA (say OAf) the tan-gent at the extremity of the corresponding ordinate AD makesan acute angle with the X-axis ; hence, fix) — + x = OA,f(x) = o. For x = OA, y =AD, and the tan-gent makes an obtuse angle with the X-axis; hence f\x) =— quantity. Hence at a maximum point, as D,f(x) passes through ofrom -j- to — direction. Similarly, to the left of the minimumpoint F the tangent makes an obtuse angle with X, while, to theright of it, the tangent makes an acute angle; hencefr(x) at aminimum point passes through o from — to -f- direction. At such a point as F, while f(x) = o, yet it does not changesign as x passes through the critical value OC,f(x) bein
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
