Inflection Point - Practice Questions & MCQ

Concavity and Point of Inflection

A point on the graph where the curve changes its concavity (upward to downward or downward to upward) is called a point of inflexion.

The tangent of the smooth curve at the point of inflection 'a' crosses the curve. Also, the second derivative $f^{\prime \prime}(x)$ changes sign at the neighbourhood of the point of inflexion. So, $\mathrm{f}^{\prime}(x)=0$ (if exists) at the point of inflexion.
But if the second derivative of the function $f(x)$ is zero at some point say $x=a$, this does not mean that this is a point of inflection.
A sufficient condition is required for a point $x=a$ to be a point of inflexion: $f(a-h)$ and $f(a+h)$ to have opposite signs in the neighbourhood of point ' $a$ '.
Consider one example
Let $\mathrm{f}(\mathrm{x})=x^4, \quad f^{\prime \prime}(x)=12 x^2$
$\Rightarrow f^{\prime \prime}(0)=0$, but at $\mathrm{x}=0$ is not a inflection point
as, $\mathrm{f}^{\prime \prime}(\mathrm{x})=12 x^2$ does not change sign in $(0-h, 0+h)$.
Consider another example
Let $y=f(x)=2 x^{1 / 3}$ here, $\mathrm{f}^{\prime}(\mathrm{x})=\frac{2}{3} \mathrm{x}^{-2 / 3}$
$f^{\prime}(x)$ is not differentiable at $x=0$ as graph has
vertical tangent at this point
But, from the graph, we observe that, at $x=0$
the curve changes its concavity.
Thus, the function doesn't need to be differentiable at the inflection point.

NOTE:

It is not necessary that at the point of inflexion, the function is continuous.