The number of real roots of the equation $\frac{(x^2+1)^3}{x^3} + \frac{x^2+1}{3x} = 0, (x \neq 0)$ is

The graph of $y = ax^2 + bx + c$ is shown. Which of the following does $NOT$ hold good?

Consider the curves given by the following quadratic functions:

$f_1(x) = 5 x^2 + 2 x + 1$	$f_2(x) = 5 x^2 + 6 x + 1$
$f_3(x) = x^2 - 7 x + 6$	$f_4(x) = 64 x^2 + 48 x + 9$

If $A_1, A_2, A_3$ and $A_4$ denote the lengths of the intercepts on the $X$-axis made by the above curves respectively,then which of the following is true?

Let $f(x) = (x - a)(x - b) - (\frac{a + b}{2})$. If $f(x) = 0$ has both non-negative roots,then the minimum value of $f(x)$ is:

Let $f(x) = x^3 + a x^2 + b x + c$ be a polynomial with integer coefficients. If the roots of $f(x)$ are integers and are in Arithmetic Progression,then '$a$' cannot take the value:

Consider a quadratic equation $ax^2 + 2bx + c = 0$ where $a, b, c$ are positive real numbers. If the equation has no real root,then which of the following is true?