A student answered the following question:

$a$ and $b$ are non-zero real numbers.

Prove that the equation $x^3+a{x}^2+b=0$ has three distinct real roots if

Here is the student's solution:

I We differentiate $y=x^3+a{x}^2+b$ to get $\frac{dy}{dx}=3x^2+2ax=x(3x+2a)$.

Solving $\frac{dy}{dx}=0$ shows that the stationary points are at $(0,b)$ and $\left(-\frac{2a}{3},b+\frac{4a^3}{27}\right)$.

II If $27b\left(b+\frac{4a^3}{27}\right)<0$, then $b$ and $b+\frac{4a^3}{27}$ must have opposite signs, and so one of the stationary points is above the $x$-axis and one is below.

III If the cubic has three distinct real roots, then one of the stationary points is above the $x$-axis and one is below.

IV Hence if $27b\left(b+\frac{4a^3}{27}\right)<0$, then the equation has three distinct real roots.

Which one of the following options best describes the student's solution?

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