(11) Max. and min. for x = 7.5 {\displaystyle x=7.5} , y = ± 5.414 {\displaystyle y=\pm 5.414} . (See example no. 10, here.)
(12) Min.: x = 1 2 {\displaystyle x={\tfrac {1}{2}}} , y = 0.25 {\displaystyle y=0.25} ; max.: x = − 1 3 {\displaystyle x=-{\tfrac {1}{3}}} , y = 1.408 {\displaystyle y=1.408} .
Exercises XI. (p. 130.)
(1) 2 x − 3 + 1 x + 4 {\displaystyle {\dfrac {2}{x-3}}+{\dfrac {1}{x+4}}} .
(2) 1 x − 1 + 2 x − 2 {\displaystyle {\dfrac {1}{x-1}}+{\dfrac {2}{x-2}}} .
(3) 2 x − 3 + 1 x + 4 {\displaystyle {\dfrac {2}{x-3}}+{\dfrac {1}{x+4}}} .
(4) 5 x − 4 − 4 x − 3 {\displaystyle {\dfrac {5}{x-4}}-{\dfrac {4}{x-3}}} .
(5) 19 13 ( 2 x + 3 ) − 22 13 ( 3 x − 2 ) {\displaystyle {\dfrac {19}{13(2x+3)}}-{\dfrac {22}{13(3x-2)}}} .
(6) 2 x − 2 + 4 x − 3 − 5 x − 4 {\displaystyle {\dfrac {2}{x-2}}+{\dfrac {4}{x-3}}-{\dfrac {5}{x-4}}} .
(7) 1 6 ( x − 1 ) + 11 15 ( x + 2 ) + 1 10 ( x − 3 ) {\displaystyle {\dfrac {1}{6(x-1)}}+{\dfrac {11}{15(x+2)}}+{\dfrac {1}{10(x-3)}}} .
(8) 7 9 ( 3 x + 1 ) + 71 63 ( 3 x − 2 ) − 5 7 ( 2 x + 1 ) {\displaystyle {\dfrac {7}{9(3x+1)}}+{\dfrac {71}{63(3x-2)}}-{\dfrac {5}{7(2x+1)}}} .
(9) 1 3 ( x − 1 ) + 2 x + 1 3 ( x 2 + x + 1 ) {\displaystyle {\dfrac {1}{3(x-1)}}+{\dfrac {2x+1}{3(x^{2}+x+1)}}} .
(10) x + 2 3 ( x + 1 ) + 1 − 2 x 3 ( x 2 − x + 1 ) {\displaystyle x+{\dfrac {2}{3(x+1)}}+{\dfrac {1-2x}{3(x^{2}-x+1)}}} .
(11) 3 ( x + 1 ) + 2 x + 1 x 2 + x + 1 {\displaystyle {\dfrac {3}{(x+1)}}+{\dfrac {2x+1}{x^{2}+x+1}}} .
(12) 1 x − 1 − 1 x − 2 + 2 ( x − 2 ) 2 {\displaystyle {\dfrac {1}{x-1}}-{\dfrac {1}{x-2}}+{\dfrac {2}{(x-2)^{2}}}} .
(13) 1 4 ( x − 1 ) − 1 4 ( x + 1 ) + 1 2 ( x + 1 ) 2 {\displaystyle {\dfrac {1}{4(x-1)}}-{\dfrac {1}{4(x+1)}}+{\dfrac {1}{2(x+1)^{2}}}} .
(14) 4 9 ( x − 1 ) − 4 9 ( x + 2 ) − 1 3 ( x + 2 ) 2 {\displaystyle {\dfrac {4}{9(x-1)}}-{\dfrac {4}{9(x+2)}}-{\dfrac {1}{3(x+2)^{2}}}} .
(15) 1 x + 2 − x − 1 x 2 + x + 1 − 1 ( x 2 + x + 1 ) 2 {\displaystyle {\dfrac {1}{x+2}}-{\dfrac {x-1}{x^{2}+x+1}}-{\dfrac {1}{(x^{2}+x+1)^{2}}}} .
(16) 5 x + 4 − 32 ( x + 4 ) 2 + 36 ( x + 4 ) 3 {\displaystyle {\dfrac {5}{x+4}}-{\dfrac {32}{(x+4)^{2}}}+{\dfrac {36}{(x+4)^{3}}}} .
(17) 7 9 ( 3 x − 2 ) 2 + 55 9 ( 3 x − 2 ) 3 + 73 9 ( 3 x − 2 ) 4 {\displaystyle {\dfrac {7}{9(3x-2)^{2}}}+{\dfrac {55}{9(3x-2)^{3}}}+{\dfrac {73}{9(3x-2)^{4}}}} .
(18) 1 6 ( x − 2 ) + 1 3 ( x − 2 ) 2 − x 6 ( x 2 + 2 x + 4 ) {\displaystyle {\dfrac {1}{6(x-2)}}+{\dfrac {1}{3(x-2)^{2}}}-{\dfrac {x}{6(x^{2}+2x+4)}}} .