Consider the equation x^6 - 5x^4 + 16x^2 - 72x +9 = 0. I calculate its 2nd derivative 30x^4 - 60x^2 + 32 which is always >0. So f '(x) is monotonic increasing for all x. So f '(x) has only 1 real root. This suggests f(x) has either 0 or 2 real roots; and this is where I am stuck. Since Descarte's rule applied on f(x) says there is 0 or 2 or 4 +ve roots and 0 or 2 -ve roots, it does not help me anyway. How can I go further?