|\^/| Maple V Release 3 (SUNY at Albany) ._|\| |/|_. Copyright (c) 1981-1994 by Waterloo Maple Software and the \ MAPLE / University of Waterloo. All rights reserved. Maple and Maple V <____ ____> are registered trademarks of Waterloo Maple Software. | Type ? for help.> f := x -> (x+2) / (3+(x^2+1)^3);

x + 2 f := x -> ------------- 2 3 3 + (x + 1)> D(f);

2 2 1 (x + 2) (x + 1) x x -> ------------- - 6 ------------------- 2 3 2 3 2 3 + (x + 1) (3 + (x + 1) )

Notice that D(f) returns the derivative of f as a function. Think of D(f) as the usual f' (i.e. f-prime the derivative of f) |

> plot(f);

The first time let Maple adjust the x and y ranges.

+A 0.5 * A * A A* A A+ A 0.4A+ A A+ A A+ A A + A 0.3 + AA A + A A + A A + A 0.2 + A A + AA A + A A + A 0.1 + AA A + A AA + AA AA + AAA ********************************-+---+---+--+-***************************** -10 -5 0 0 5 10 > f(0); 1/2 > # Do you see that the global max of f is at an x a bit bigger than 0 ? > plot(f(x), x=-5..5, y=-0.1..0.1); A 0.1 + A AA + A A + A A + A A + A A 0.05 + A A + A A + A A + AA AA + AAA -+--+--+--+--+--+--+--****--+--+--+--+--+--+--+--+--+--+-****************** AAAAAAAAAAAAAAAAAAAAAA + -4 -2 0 0 2 4 + + + -0.05 + + + + + -0.1 + > d := simplify(D(f)(x)); 6 4 2 5 3 - 4 + 5 x + 9 x + 3 x + 12 x + 24 x + 12 x d := - ----------------------------------------------- 6 4 2 2 (4 + x + 3 x + 3 x ) > fsolve(numer(d)); -2.485165927, .2700964050 > # Let's grab this list of two numbers for later use. > s := "; s := -2.485165927, .2700964050 > # A quick check... > D(f)(s[2]); -9 .1*10 > # VERY close to 0 as it should. Why isn't exactly zero? > # Now the inflection points... > # We use now the command "diff" instead of "D".... See the difference? > diff(d,x); 5 3 4 2 30 x + 36 x + 6 x + 60 x + 72 x + 12 - ---------------------------------------- 6 4 2 2 (4 + x + 3 x + 3 x ) 6 4 2 5 3 5 3 (- 4 + 5 x + 9 x + 3 x + 12 x + 24 x + 12 x) (6 x + 12 x + 6 x) + 2 ---------------------------------------------------------------------- 6 4 2 3 (4 + x + 3 x + 3 x ) > # Use 'simplify' to combine the two fractions into a single one. > simplify("); 5 11 9 7 3 4 10 8 6 (- 19 x + 5 x + 17 x + 18 x - 37 x - 12 x + 14 x + 14 x + 54 x 6 2 / 6 4 2 3 + 76 x - 30 x - 8) / (4 + x + 3 x + 3 x ) / > fsolve(numer(")); -2.964270928, -.5472819999, .8702050673 > # So these are the x locations of the three inflection points. > # Had we used 'solve' instead of 'fsolve' we would have obtained the > # exact values but usually at the expense of complicated expresions. > # We can now adjust the x and y ranges to see what's going on > # to the left of 0. > plot(f(x), x=-4..0, y=-0.01..0.01); A 0.01 A + A + A + A + AA 0.005 A + AA + AA + A + +---+--+---+---+--+---+---+---+--+---**--+--+---+---+---+--+---+---+--+---+ AAAAAAAAAAAAAAAAA AAA |0 -4 -3AAAAAAAAAAAAAAA -2 -1 0 + + + -0.005 + + + + -0.01 > # Do you see the inflection pt. around -3 ? > # Also, do you see the global min of f.. at around s[1] = -2.5 ? >

Carlos Rodriguez <carlos@math.albany.edu> Last modified: Wed Sep 4 18:31:06 EDT 1996