(%i1) "Funktionsuntersuchung"$


(%i2) f(x) := (2-x)/(x^2-2);
					2 - x
(%o2) 			        f(x) := ------
					 2
					x  - 2

(%i3) "Schnittpunkt mit y-Achse"$


(%i4) f(0);
(%o4) 				      - 1

(%i5) "Nullstellen"$


(%i6) solve((2-x)/(x^2-2)= 0, x);
(%o6) 				    [x = 2]

(%i7) ev(%,numer);
(%o7) 				    [x = 2]

(%i8) "1. Ableitung"$


(%i9) diff(f(x),x) ;
				1      2 (2 - x) x
(%o9) 			    - ------ - -----------
			       2	  2     2
			      x  - 2    (x  - 2)

(%i10) sqfr(%);
				  2
				 x  - 4 x + 2
(%o10) 				 ------------
				    2	  2
				  (x  - 2)

(%i11) "Extremstellen"$


(%i12) solve( %th(2) = 0,x);
(%o12) 		      [x = 2 - sqrt(2), x = sqrt(2) + 2]

(%i13) ev(%,numer);
(%o13) 		 [x = 0.5857864376269, x = 3.414213562373095]

(%i14) "2. Ableitung"$


(%i15) diff(f(x),x,2) ;
							2
			4 x	 2 (2 - x)   8 (2 - x) x
(%o15) 		     --------- - --------- + ------------
		       2     2	   2	 2      2     3
		     (x  - 2)	 (x  - 2)     (x  - 2)

(%i16) sqfr(%);
				 3      2
			     2 (x  - 6 x  + 6 x - 4)
(%o16) 			   - -----------------------
				      2	    3
				    (x  - 2)

(%i17) "Wendestellen"$


(%i18) solve( %th(2) = 0,x);
	       sqrt(3) %i   1
	    2 (---------- - -)
		   2	    2	   3/2	   1/3	  sqrt(3) %i   1
(%o18) [x = ------------------ + (2    + 4)    (- ---------- - -) + 2, 
		3/2     1/3			      2	       2
	      (2    + 4)
					  sqrt(3) %i   1
				     2 (- ---------- - -)
      3/2     1/3  sqrt(3) %i   1	      2	       2
x = (2	  + 4)	  (---------- - -) + -------------------- + 2, 
		       2        2	  3/2	  1/3
					(2    + 4)
      3/2     1/3	  2
x = (2	  + 4)	  + ------------- + 2]
		      3/2     1/3
		    (2	  + 4)

(%i19) ev(%,numer);
(%o19) [x = 1.054200220504741 (0.86602540378444 %i - 0.5)
 + 1.8971728150867 (- 0.86602540378444 %i - 0.5) + 2, 
x = 1.8971728150867 (0.86602540378444 %i - 0.5)
 + 1.054200220504741 (- 0.86602540378444 %i - 0.5) + 2, x = 4.951373035591441]

(%i20) "3. Ableitung"$


(%i21) diff(f(x),x,3) ;
			       2			        3
		 6	   24 x	     24 (2 - x) x   48 (2 - x) x
(%o21) 	     --------- - --------- + ------------ - -------------
	       2     2	   2	 3      2     3	        2     4
	     (x  - 2)	 (x  - 2)     (x  - 2)	      (x  - 2)

(%i22) sqfr(%);
			   4	  3	  2
		       6 (x  - 8 x  + 12 x  - 16 x + 4)
(%o22) 		       --------------------------------
				    2	  4
				  (x  - 2)

(%i23) "Stammfunktion"$


(%i24) integrate(f(x), x);
				    3/2
			     2 x - 2
			 log(----------)
				    3/2	        2
			     2 x + 2	   log(x  - 2)
(%o24) 			 --------------- - -----------
			     sqrt(2)	        2

(%i25) sqfr(%);
				 3/2
			  2 x - 2		     2
		    2 log(----------) - sqrt(2) log(x  - 2)
				 3/2
			  2 x + 2
(%o25) 		    ---------------------------------------
				   2 sqrt(2)

(%i26) functions;
(%o26) 				    [f(x)]

(%i27) plot2d([f(x)],[x,-4,4],[y,-10,10],[gnuplot_preamble,"set grid; set zeroaxis linetype -1;"]);
(%o27)