. Applied calculus; principles and applications . - 6^. 15. Show that e is a minimum of x/log x. 16. Show that 1/ne is a maximum of log x/x^. 17. Show that e^^^ is a maximum of x^^. 18. Show that 1 is a maximum of 2 tan ^ — tan^ d. 19. Find the maximum value of tan^ x — tan^ x/4, the anglesbeing taken in the first quadrant. Ans. tan~^ f. 20. Show that 2 is a maximum ordinate and —26 is a minimumordinate of the curve y = x^ — 5x*-^5x^-\-l. EXERCISE XI 127 PROBLEMS IN MAXIMA AND MINIMA. 1. Find the maximum rectangle that can be inscribed in a circle ofradius a. Let 2 x = base and 2 y = altitude;

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. Applied calculus; principles and applications . - 6^. 15. Show that e is a minimum of x/log x. 16. Show that 1/ne is a maximum of log x/x^. 17. Show that e^^^ is a maximum of x^^. 18. Show that 1 is a maximum of 2 tan ^ — tan^ d. 19. Find the maximum value of tan^ x — tan^ x/4, the anglesbeing taken in the first quadrant. Ans. tan~^ f. 20. Show that 2 is a maximum ordinate and —26 is a minimumordinate of the curve y = x^ — 5x*-^5x^-\-l. EXERCISE XI 127 PROBLEMS IN MAXIMA AND MINIMA. 1. Find the maximum rectangle that can be inscribed in a circle ofradius a. Let 2 x = base and 2 y = altitude; thenarea A = 4: xy =4:X Va^ — x^.. Take / (x) = x^ (a^ - x^) = a^x^ - x^ [by Art. 87]; / (x) =2aH-4:X^ = 2x (a^ -2x^) = 0; .-. x = 0, a/V2;J {x) =2a^-12x^; f (0) = 2 a^; /. / (0) = 0 is ) = 2o? - 6a2 = -4a2; .-. /(a/V2) = A = 4 V^ = 2 a2 is the area of the maximum rectangle, which is a — By Geometry without the Calculus method: A =2ay = 2aVa^ - x-]j^o = 2a2, since the radical quantity is evidently greatest when x = 0.

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