Inflection Point: Definition, Graph and Example

Concavity and Point of Inflection is an important concepts in calculus. It is useful in understanding the relationship between curves and their slopes. The monotonic function is either increasing or decreasing. These concepts of monotonicity have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

In this article, we will cover the concept of Monotonicity. This topic falls under the broader category of Calculus, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE.

Concavity and Point of Inflection:

For concavity:

If $f^{\prime \prime}(x)>0$ in the interval $(a, b)$ then shape of $\mathrm{f}(\mathrm{x})$ in interval $(a, b)$ is concave when observed from upwards or convex down.

For convexity:

If $f^{\prime \prime}(x)<0$ in the interval $(a, b)$ then it is convex upward or concave down.

Point of inflection

If at x = a the shape of the curve changes from concave to convex or from convex to concave. Then $x = a$ is known as the point of inflection and $f^{\prime \prime}(x)=0 \quad$ at $\quad x=a$

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

If f '(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. In fact, such a point is called a point of inflection

The tangent of the smooth curve at the point of inflection ‘a’ crosses the curve. Also, the second derivative f’’(x) changes sign at neighbourhood of point of inflection. So, $f’’(x) = 0$ (if exists) at the point of inflection.

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 inflection: f(a - h) and f(a + h) to have an opposite sign in the neighborhood of point ‘$a$’.

We now finally come to the point of inflection consider $f(x)=x^3$ again $f^{\prime}(0)=f^{\prime \prime}(0)=0$ $f^{\prime \prime}(x)<0$ for $x<0$and $f(x)>0$ for $x>0$

$\Rightarrow f^{\prime \prime}(x)$ changes sign as x crosses 0 .

$\Rightarrow f(x)$ changes the nature of its concavity as x crosses 0 such a point is called a point of inflection, a point at which the concavity of the graph changes.

Condition for inflexion

1. f''(x) changes sign x passes through the point a.

2. only those values of x for which f''(x) change signs are points of inflexion.

ex : $\quad f^{\prime \prime}(x)=x^2(x-2), \quad f^{\prime \prime}(x)=0$

at $x=0$ and $x=2$ but only $x=2$ point of in flexion because at $x=2 f^{\prime \prime}(x)$ changes sign only

Consider one examples

Consider another example

Let $y=f(x)=2 x^{1 / 3}$

here, $f^{\prime}(x)=\frac{2}{3} x^{-2 / 3}$

$f^{\prime}(x)$ is not differentiable at $x=0$ as graph has vertical tangent at this point

But, from the graph we observe that, at $x=0$ curve changes its concavity.

Thus, it is not necessary that function is differentiable at inflection point.

NOTE:

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

Solved Examples Based on Concavity and Point of Inflection:

Example 1: $f(x)=x^3-3 x^2$ has concaity upwards in the interval

1) $(1, \infty)$

2) $(-\infty, \infty)$

3) $(-1, \infty)$

4) $(-\infty, 1)$

Solution

As we have learned

Concavity, Convexity, of a function -

If $f^{\prime \prime}(x)>0$ in the interval ${{(a, b)}}$ then shape of $\mathrm{f}(\mathrm{x})$ in interval $(a, b)$ is concave when observed from upwards or convex down.

For convexity:

If $f^{\prime \prime}(x)<0$ in the interval $(a, b)$ then it is convex upward or concave down.

- wherein

$

f^{\prime}(x)=3 x^2-6 x \Rightarrow f^{\prime \prime}(x)=6 x-6=6(x-1)

$

for concave up $\rightarrow f^{\prime \prime}(x)>0 \Rightarrow 6(x-1)>0 \Rightarrow x>1$

Example 2: $f(x)=x^4+2 x^3+6 x^2+12 x$ has concavity upwards only in the interval

1) $(1, \infty)$

2) $(-\infty, 0)$

3) $(0, \infty)$

4) $(-\infty, \infty)$

Solution

As we have learned

Concavity, Convexity, of a function -

For concavity:

If $f^{\prime \prime}(x)>0$ in the interval $(a, b)$ then shape of $\mathrm{f}(\mathrm{x})$ in interval $(a, b)$ is concave when observed from upwards or convex down.

For convexity:

If $f^{\prime \prime}(x)<0$ in the interval $(a, b)$ then it is convex upward or concave down.

- wherein

$

\begin{aligned}

& f^{\prime}(x)=4 x^3+6 x^2+12 x+12 \\

& f^{\prime \prime}(x)=12 x^2+12 x+12=12\left(x^2+x+1\right)>0 \forall n \in R \\

& \because f^{\prime \prime}(x)>0 \ldots \forall x \in R

\end{aligned}

$

$\Rightarrow f(x)$ is concave up $\forall n \in R$

Example 3: Which of the following curves shows $x = a$ as point of inflexion ?

1)

2)

3)

4)

Solution

As we have learned

Point of inflexion -

If at x = a the shape of the curve changes from concave to convex from convex to concave then at $x = a$ is known as the point of inflexion and

$f^{\prime \prime}(x)=0 \quad$ at $x=a$

In (A), concavity remains downward in left and right both of '$a$'

In (B), concavity changes from downward to upward about '$a$'

In (C), concavity remains downward about '$a$'

In (D), concavity remains upwards about '$a$'

Example 4: Which of the following is not a point of inflexion in $y=\sin x$

1) $\frac{\pi}{2}$

2) $\pi$

3) $2 \pi$

4) $3 \pi$

Solution

As we have learned

Point of inflection -

If at x = a the shape of the curve changes from concave to convex from convex to concave then at x = a is known as the point of inflection and

$f^{\prime \prime}(x)=0 \quad$ at $\quad x=a$

From the graph we see, concavity remains downward about $x=\pi / 2$ but about $\pi, 2 \pi, 3 \pi$ it changes, so $x=\pi / 2$ is not a point of inflection

Example 5: Point of inflection for $f(x)=x^3$ is at $x$ equals

1) -1

2) $\frac{1}{2}$

3) 0

4) 1

Solution:

As we have learned

Condition for point of inflection-

1. f''(x) changes sign x passes through the point a.

2. only those values of x for f''(x) change signs are points of inflection.

$

\text { e.g. : } f^{\prime \prime}(x)=x^2(x-2), \quad f^{\prime \prime}(x)=0

$

at $x=0$ and $x=2$ but only $x=2$ point of inflexion because at $x=2 f^{\prime \prime}(x)$ changes sign only

$f^{\prime}(x)=3 x^2 \Rightarrow f^{\prime \prime}(x)=6 x$ which will change its sign at $x= 0$, so concavity will change at $x=0$, so $x=0$ is a point of inflection.

Hence, the answer is the option 3.

Summary

Concavity is an important part of mathematics. The point of inflection is useful to find the concavity and convexity of the curve. This concept is used in many parts of real life like in physics and chemistry.