Use differentiation to determine the point(s) of inflection of the function: y = x³ + 2x² − 5x − 6

The given function is $y = x^3 + 2x^2 - 5x - 6$.
Differentiate $y$ with respect to $x$:

Differentiate $\frac{dy}{dx}$ with respect to $x$:

Points of inflection occur where the concavity changes, which typically happens when $\frac{d^2y}{dx^2} = 0$.

We can check the sign of $y''$ for values of $x$ less than and greater than $-\frac{2}{3}$.
If $x < -\frac{2}{3}$ (e.g., $x = -1$): $y'' = 6(-1) + 4 = -6 + 4 = -2 < 0$ (concave down).
If $x > -\frac{2}{3}$ (e.g., $x = 0$): $y'' = 6(0) + 4 = 4 > 0$ (concave up).
Since the sign of $y''$ changes at $x = -\frac{2}{3}$, this is indeed an inflection point.

Substitute $x = -\frac{2}{3}$ into $y = x^3 + 2x^2 - 5x - 6$:

To combine these, find a common denominator, which is 27:

The point of inflection is $\left(-\frac{2}{3}, -\frac{56}{27}\right)$.