This worksheet is intended to illustrate what can happen in Maple if you graph a function without specifying "real" output.

The worksheet also includes an absolute maximum/minimum over an interval example.

>	with(plots):

Warning, the name changecoords has been redefined

>	f:=x*(4-x^2)^(1/3);

>	diff(f,x);

>	fPrime:=diff(f,x):

>	solve({fPrime=0});

>	fGraph:=plot(f,x=-4..4,y=-4..4,thickness=2):

>	display(fGraph);

>	eval(f,x=sqrt(5));

>	(-8)^(1/3);

>

>	solve({x^3+8=0});

What happened above is that Maple interpreted the cube root of a negative number as not being real and therefore the function was not defined outside the interval [-2,2].

Two of the three cube roots of a negative number would not be real (see above example).  We can specify only real output for the function using RealDomain.

>	with(RealDomain):

Warning, these protected names have been redefined and unprotected: Im, Re, ^, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, cos, cosh, cot, coth, csc, csch, eval, exp, expand, limit, ln, log, sec, sech, signum, simplify, sin, sinh, solve, sqrt, surd, tan, tanh

>	f2:=x*(4-x^2)^(1/3);

>	f2Graph:=plot(f2,x=-4..4,y=-4..4,thickness=2):

>	display(f2Graph);

>	(-8)^(1/3);

>	eval(f2,x=sqrt(5));

>	f2Prime:=diff(f2,x);

>	eval(f2Prime,x=1);

>

evalf(%);

>	eval(f2Prime,x=sqrt(12/5));

>	eval(f2Prime,x=2);

Error, (in assuming) when calling 'signum'. Received: 'signum is not differentiable at 0'

The function is not differentiable at -2 and 2 (the tangent lines would be vertical).

The example below investigates finding the absolute maximum and absolute minimum value of a function (f3) over the closed interval [-4,4].

>	f3:=abs(x^3-12*x);

>	f3Graph:=plot(f3,x=-4..4,y=-40..20,thickness=2):

>	display(f3Graph);

>	f3Prime:=diff(f3,x);

• The derivative of abs  is denoted by abs(1, x) . This is signum(x)  for all non-0 real numbers, and is undefined otherwise.

>	solve([f3Prime=0],[x]);

>	eval(f3Prime,x=2*sqrt(3));

Error, (in assuming) when calling 'simpl/abs'. Received: 'abs is not differentiable at 0'

Notice that Maple considered f3Prime to be zero at three values of x where it was actually undefined.  Since the function is defined at those values of x they are still critical values.  To answer the question we need to evaluate f3 at all critical values in the interval [-4,4] and also at -4 and 4.

>	eval(f3,x=-4);

>	eval(f3,x=-2*sqrt(3));

>	eval(f3,x=-2);

>	eval(f3,x=-0);

>	eval(f3,x=2);

>	eval(f3,x=2*sqrt(3));

>

The maximum value of the function over the interval is 16 (and occurs at three different values of x) and the minimum value of the function over the interval is 0 (and also occurs at three places).