Hello, Punch!

Your work is almost correct.

And be sure to *answer the question*.

Find the *smallest integer value* of $\displaystyle b$ for which $\displaystyle 3x^2+bx+2$ is positive for all values of $\displaystyle x.$

My attempt: .$\displaystyle b^2\:<\:24 \qquad{\color{blue}\Longrightarrow}\qquad {\color{blue}|b| \:<\:\sqrt{24}}$

However, the answer is $\displaystyle -4$

We have: .$\displaystyle -\sqrt{24} \:<\:b\:<\:\sqrt{24}$

. . . . . . . . $\displaystyle -4.9 \:<\:b\:<\:4.9 $

So $\displaystyle b$ is *between* -4.9 and +4.9

Code:

↓
- - + o * * * * * * * * * * * * * * * * * o + - -
-5 -4 -3 -2 -1 0 1 2 3 4 5

So the least integer value of $\displaystyle b$ is -4.

Edit: ah, Tonio beat me to it . . . *sigh*

.