Number of distinct real roots The equation $x^6 − 5x^4 + 16x^2 − 72x+ 9 = 0$ has
(A) exactly two distinct real roots
(B) exactly three distinct real roots
(C) exactly four distinct real roots
(D) six distinct real roots.
$f(0)>0$ and $f(1)<0$ and $f(3)>0$, so there should be an odd no of roots between 0 and 1 and 1 and 3 but exactly how many?
 A: The second derivative of $f(x)=x^6 − 5x^4 + 16x^2 − 72x+ 9 = 0$ is
$$f''(x)=30x^4-60x^2+32,$$
which is quadratic in $x^2$ having negative discriminant $-240$ and so is never zero. Since it's positive at $x=0$ we have $f''(x)>0$ for all real $x$, i.e. $f(x)$ is concave up on $\mathbb{R}$. This means $f(x)$ can have at most two real zeros, and you have already shown at least two.
A: I haven’t tried it, but if you derive your equation twice, you can substitute $z = x^2$, solve for $z$ and get the inflection points of your equation from which you should be able to derive the number of zeros in the original equation.
Let’s see:
$$\frac{d^2}{dx^2}x^6−5x^4+16x^2−72x+9 = 30 x^4 - 60 x^2 + 32$$
Substituting $z = x^2$ and dividing by $2$, for the discriminant of $15z^2 - 30 z + 16$ you get:
$900 - 4·15·16 = 900·(1 - \frac{32}{30}) < 0$, so you have no inflection points.
What does this tell us? Essentially that $x^6−5x^4+16x^2−72x+9 - y$ can only have two roots for if it had more than two, then it would have more than one extreme point and then it would have at least one inflection point by Rolle’s theorem.
