/*
** 1-Variable Scalar-Valued Function
** f: R -> R
*/
fn f(x)=ln(x)-x^2;   @ 1-variable scalar-valued function @
fn f1(x)=1/x-2*x;    @ 1st analytic derivative of f(x) @
fn f2(x)=-1/(x^2)-2; @ 2nd analytic derivative of f(x) @

v=4;            			@ function evaluation at point v=4 @
print f(v);                 @ f(4) @
print f1(v);                @ f1(4) @
print f2(v);                @ f2(4) @

/*
Numeric Derivatives
*/
print gradp(&f,v);			@ 1st numeric derivative of f(x) @
print hessp(&f,v);			@ 2nd numeric derivative of f(x) @

@ find maximum of f(x) at x=sqrt(0.5): @
@ check: f1=0 and f2<0 @

/*
Using graph to find the maximum:
Make sure that your GAUSS is configured correctly with your
equipment: monitor and printer in particular. Modify the file
PQGRUN.CFG if necessary, according to the instruction in the file.
*/
library pgraph;
graphset;
x=seqa(0.0001,0.1,40);
y=f(x);
call xtics(0,4,0.5,0);
call ytics(-15,0,1,0);
call xy(x,y);

end;