Which of the following equations has the greatest number of real solutions?(math subject GRE exam 1268 Q.17) Could anyone give me a hint for the most rapid way for solving this question (solving it in exactly 2.5 minutes)?
Thanks!



 A: I doubt this is really doable in 2'30" (unless you are already in full concentration mode).
You spot at once that D has no solution (knowing that $e^x>x$) and C (first degree) has one. Rewrite E as $\cos x=e^{x^2}\ge1$ and it has exactly one solution ($x=0$).
B is quadratic and potentially has two solutions. By checking the discriminant, $76$, it indeed has them.
A is cubic and potentially has three roots. But the derivative $3x^2+1$ is positive so that there are no extrema and a single root.
A: I tried in 2.5. minutes, so more a guess than anything else really, for somebody of my limited skills.
So, $E$ has one real solution for $x=0$, as the Gaussian is bounded above by $1$ and the secant is bounded below by the same value, a graphical approach more than rigorous.
$A$ also has one real solution due to, I hope, a quick derivative check.
Same for $C$ and $D$ (due to linearity and and easy exponential inequality respectively).
My best guess is then $B$, which I verified it has two by checking the discriminant.
