This is a Test of Mathematics Solution Subjective 65 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

## Problem

Show that for all real x, the expression + bx + C ( where a, b, c are real constants with a > 0), has the minimum value . Also find the value of x for which this minimum value is attained.

## solution:

f (x) + bx + c

Now minimum derivative = 0 & 2nd order derivative > 0.

= 2ax + b

Or =2a

Now 2a> so 2nd order derivative > 0 so = 2.

So minimum occurs when

= 0 or 2ax + b = 0

or 2ax = -b

or x = (ans)

At x =

+ bx + c

= + + c

= (proved)