Find all the points of local maxima and local minima of the function $f$ given by $f(x) = 2x^3 - 6x^2 + 6x + 5.$

No points of local maxima or minima

The correct answer is Option (3) → No points of local maxima or minima ##

We have

$f(x)= 2x^3 - 6x^2 + 6x + 5$

$\text{or } \quad f'(x) = 6x^2 - 12x + 6 = 6(x - 1)^2$

$\text{or } \quad f'(x) = 0 \quad \text{at } x = 1$

Thus, $x = 1$ is the only critical point of $f$. We shall now examine this point for local maxima and/or local minima of $f$. Observe that $f'(x) \geq 0$, for all $x \in \mathbf{R}$ and in particular $f'(x) > 0$, for values close to $1$ and to the left and to the right of $1$. Therefore, by first derivative test, the point $x = 1$ is neither a point of local maxima nor a point of local minima. Hence $x = 1$ is a point of inflexion.