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?

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.
