If \(a\) and \(b\) are the maximum and minimum values of the quadratic expressions \(1-2 x-5 x^2\) and…
If \(a\) and \(b\) are the maximum and minimum values of the quadratic expressions \(1-2 x-5 x^2\) and \(x^2-2 x+5\) respectively, then the set of all values of \(x\) for which the expression \(5 a x^2+b x+7\) is positive, is
\((a, b)\)
\((-\infty, 7)\)
\((5, \infty)\)
\((-\infty, \infty)\)
Solution
The maximum value of the expression
\(\begin{aligned}
& 1-2 x-5 x^2 \\
& a=-\frac{4+20}{4(-5)}=\frac{6}{5}
\end{aligned}\)
and minimum value of the expression \(x^2-2 x+5\),
\(b=-\frac{4-20}{4}=4\)
Now, the given quadratic expression \(5 a x^2+b x+7\) at \(a=\frac{6}{5}\) and \(b=4\) must be positive, then
\(6 x^2+4 x+7 > 0\)
\(\because\) Discriminant \(D=16-4(6)(7) < 0\) and coefficient of \(x^2\) term is positive.
\(\therefore \quad 6 x^2+4 x+7 > 0, \forall x \in R=(-\infty, \infty) \text {. }\)
Hence, option (d) is correct.