The value of a so that the sum of the cubes of the roots of the equation `x^2 ax+ (2a -3)=0` assumes the minimum vlaue's

Options

A. a=1
B. a=3
C. a=0
D. non of these

Answer

B

Explanation

Let `alpha and beta ` be the roots of the given equation Then
`alpha + beta =a and alpha beta=2a -3`
Let `S=alpha^3 + beta^3` Then
`S= (alpha + beta)^3-3 alpha beta (alpha + beta) `
` rArr S=a^3-3a(2a-3)`
`rarr S=a^3 -6a^2 + 9a `
`rArr (dS)/(da)=3a^2-12a + 9= 3 (a^2-4a +3)=3(a-1)(a-3)`
Clearly `(ds)/(da)=0` when a=1,3
The changes in the signs of ds/da are shown in Fig.5

Clearly ,S is minimum at a=3
Recorder to find the local maximum / minimum values of a function there is alsol an alternative methaod known as higher order derivative test as stated below .