(a) Find the smallest value of the integer \(a\)for which \(a x^2+5 x+2\) is positive for all values of \(x\).
(b) Find the smallest value of the integer \(b\) for which \(-5 x^2+b x-2\) is negative for all values of \(x\).
(a) Since \(a x^2+5 x+2\) is positive for all values of \(x\), \(a>0\) and \(y=a x^2+5 x+2\) has no \(x\)-intercepts.
Positive for all values of \(x\) means that if we were to draw a graph of the quadratic function, all the \(y \) coordinates of the points on the graph will be positive. In other words, the graph will not intersect the x axis, since the coordinate of any point on the x axis is \(0\). Also the graph has a minimum turning point
The discriminant tells us whether the graph of a quadratic function intersects the x axis. Since the graph does not intersect the x axis, discriminant \(<0\)
$$ \begin{aligned} (5)^2-4(a)(2) & <0 \\ 25-8 a & <0 \\ a & >3 \frac{1}{8}
\end{aligned}
$$
the smallest integer value of \(a\) is 4 .
(b) Since \(-5 x^2+b x-2\) is negative for all values of \(x\), \(y=-5 x^2+b x-2\) has no \(x\)-intercepts and has a maximum turning point.
Discriminant \(<0\)
\( b^2-4(-5)(-2)<0\)
\(b^2-40<0\)
\(b^2-(\sqrt{40})^2<0\)
\(-\sqrt{40}<b<\sqrt{40}\)
the smallest integer value of \(b\) is \(\mathbf{- 6}\).