यदि फलन f(x), जहाँ `f(x)={{:(3ax + b",","यदि",x gt 1),(11",","यदि",x = 1),(5ax-2b",","यदि",x lt 1):} x = 1` पर संतत हो, तो a और b का मान ज्ञात कीजिए ।

Explanation

(i) `f(1)=11`
(ii) R.H.L. `=underset(x rarr 1^(+))(lim)f(x)=underset(h rarr 0)(lim)f(1+h)`
`=underset(h rarr 0)(lim)[3a(1+h)+b]," "[because x = 1 + h gt 1]`
`=underset(h rarr 0)(lim)(3a + 3ah + b)`
`=3a + b + 3a xx 0`
`rArr" ""R.H.L."=3a + b`
(iii) L.H.L. `=underset(x rarr 1^(-))(lim)f(x)=underset(h rarr 0)(lim)f(1-h)`
`=underset(h rarr 0)(lim)[5a(1-h)-2b]," "[because x = 1 - h lt 1]`
`=underset(h rarr 0)(lim)5a - 5ah - 2b`
`=5a - 5a xx 0 - 2b`
`rArr" ""L.H.L." =5a - 2b`
चूँकि फलन f(x) बिन्दु x = 1 पर संतत है ।
`therefore" ""R.H.L.=L.H.L."=f(1)`
`rArr" "3a + b = 5a - 2b = 11`
`rArr" "3a + b = 11" "...(1)`
और `5a - 2b = 11" "...(2)`
समी. (1) को 2 से गुणा करके समी. (2) से जोड़ने पर,
`{:(6a+2b = 22),(ul(5a - 2b = 11)),(11a = 33):}`
`rArr" "a = 3`
समी. (1) में a = 3 रखने पर,
`3 xx 3 + b = 11 rArr b = 11 - 9 = 2`
अत: a = 3 और b = 2