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Q.
If a and b are non-zero roots of $x^2 + ax + b = 0$ then the least value of $x^2 + ax + b$ is
Application of Derivatives
Solution:
As given a and b are the roots of the equation
$x^2 + ax + b = 0$
$\Rightarrow $ sum of roots, a + b = - a
$\Rightarrow \, b = - 2a $ ...(1)
and product of roots, ab = b
$\Rightarrow \, ab - b = 0$
$\Rightarrow \, b (a - 1) = 0$
if b = 0 then a = 0
if b $\neq$ 0 then a = 1 and b = - 2
so, the expression will be,
$f (x) = x^2 + x - 2$
$ = x^{2} + 2. \frac{1}{2} x + \left(\frac{1}{2}\right)^{2} - \left(\frac{1}{2}\right)^{2} - 2$
$ \Rightarrow f\left(x\right) = \left(x + \frac{1}{2}\right)^{2} - \frac{9}{4}$
So, f (x) will be minimum, if $ \left(x + \frac{1}{2}\right)^{2} = 0$
i.e. when $ x = -\frac{1}{2}$
$ \Rightarrow $ minimum value of function $ = - \frac{9}{4} $