Cubical Parabola has Point of Inflection

Theorem

The cubical parabola has a point of inflection.

When the equation for the cubical parabola is in the form $y = a x^3$, this point of inflection is the origin.

Proof

\(\ds y\)

\(=\)

\(\ds a x^3\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds 3 a x^2\)

Derivative of Power

By definition, there is a point of inflection at a point where $x = \xi$ if and only if the first derivative has either a local maximum or a local minimum at $\xi$.

We note that $3 a x^2 \ge 0$ is the equation for the parabola.

From Square of Non-Zero Real Number is Strictly Positive, we note that:

$3 a x^2 = 0 \iff x = 0$

and:

$\forall x \in \R: x \ne 0 \implies x^2 > 0$

So:

if $a > 0$ then $3 a x^2$ has a local minimum at $x = 0$

if $a < 0$ then $3 a x^2$ has a local maximum at $x = 0$.

Hence the result.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cubical parabola
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cubical parabola