函數(shù)eq\r(50x)+eq\r(121y)=13的性質(zhì)與圖像※.函數(shù)的定義域∵eq\r(121y)=13-eq\r(50x)≥0,∴eq\r(50x)≤13,即:0≤x≤eq\f(169,50).則函數(shù)的定義域?yàn)椋篬0,eq\f(169,50)].※.函數(shù)的單調(diào)性對(duì)方程兩邊同時(shí)對(duì)自變量x求導(dǎo)，得：eq\f(50,2eq\r(50x))+eq\f(121y',2eq\r(121y))=0,eq\f(121y',eq\r(121y))=-eq\f(50,eq\r(50x)),y'=-eq\f(50,121)*eq\f(eq\r(121y),eq\r(50x))<0即函數(shù)y在定義域上為單調(diào)減函數(shù)。當(dāng)x=0時(shí)，ymax=eq\f(169,121);當(dāng)x=eq\f(169,50)時(shí)，ymin=0。則函數(shù)的值域?yàn)椋篬0,eq\f(169,121)]。本題也可通過復(fù)合函數(shù)性質(zhì)來判斷函數(shù)單調(diào)性，因?yàn)閑q\r(50x)+eq\r(121y)=13，所以eq\r(121y)=13-eq\r(50x)，又因?yàn)楹瘮?shù)y1=eq\r(50x)為增函數(shù)，則取負(fù)號(hào)后為減函數(shù)，即f(eq\r(121y))為減函數(shù)。※.函數(shù)的凸凹性∵y'=-eq\f(50,121)*eq\f(eq\r(121y),eq\r(50x))=-eq\f(50,121)*eq\f(eq\r(121),eq\r(50))*eq\f(eq\r(y),eq\r(x)).∴y＂=-eq\f(50,121)*eq\f(eq\r(121),eq\r(50))*(eq\f(eq\r(y),eq\r(x)))'.=-eq\f(50,121)*eq\f(eq\r(121),eq\r(50))*eq\f(\f(\r(x),2\r(y))y'-\f(\r(y),2\r(x)),x)=-eq\f(50,121)*eq\f(eq\r(121),eq\r(50))*eq\f(-\f(\r(x),2\r(y))(eq\f(50,121)*eq\f(eq\r(121),eq\r(50))*eq\f(eq\r(y),eq\r(x)))-\f(\r(y),2\r(x)),x)=eq\f(50,121)*eq\f(eq\r(121),eq\r(50))*eq\f(\f(\r(x),2\r(y))(eq\f(50,121)*eq\f(eq\r(121),eq\r(50))*eq\f(eq\r(y),eq\r(x)))+\f(\r(y),2\r(x)),x)>0，所以函數(shù)y在定義域上為凹函數(shù)?！?函數(shù)的五點(diǎn)圖x00.841.692.533.38eq\r(50x)06.489.1911.21313-eq\r(50x)136.523.811.80y1.3960.3510.110.020※.函數(shù)的示意圖y（0,1.396）(0.84,0.351)(1