Show that the minimum value of `(x+a)(x+b)//(x+c)dotw h e r ea > c ,b > c ,`is `(sqrt(a-c)+sqrt(-c))^2`for real values of `x >-cdot`
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Given expression is `(x + a) (x + b) //(x + c).`
Let `x + c + y `. Then
`((x + a) (x + a))/((x + c)) = ((y+ (a – c))(y+(b-c)))/(y)`
` = (y^(2)+ [(a – c)+(b -c)]y+(b-c)(b – c))/(y)`
` = y+((a – c)+(b -c))/y+(b-c)+(b – c)`
`=[sqrt(y)-sqrt((a-c)(b-c))/(y)]^(2) + [sqrt(a-c)+sqrt(b-c)]^(2)`
` =ge[sqrt(a – c)+sqrt(b -c)]^(2)`
Hence, the least value is `[sqrt(a – c)+sqrt(b -c)]^(2)`