IB Analysis and Approaches HL · Differential calculus · Section 10 of 14

Concavity and points of inflexion

3 worked examples · By Álvaro López

Key idea

The first derivative tells you whether a function is rising or falling; the second derivative tells you how the graph curves. With it you locate points of inflexion, classify maxima and minima and sketch graphs accurately. In IB it appears in Paper 1 and Paper 2 questions, often with the graph of $f^{'}$ or $f^{''}$ given as data.

Concavity

A function is concave up on an interval if its graph lies above its tangents: it curves like a smile. It is concave down if it lies below them: it curves like a frown.

Look at how the gradients of the tangents change. On a curve that is concave up, the gradients increase from left to right, that is, $f^{'}$ is increasing. On a curve that is concave down, the gradients decrease, and $f^{'}$ is decreasing. Since $f^{''}$ is the derivative of $f^{'}$ , the sign of $f^{''}$ decides.

Points of inflexion

That $f^{''} (x_{0}) = 0$ is not enough. The function $f (x) = x^{4}$ has $f^{''} (0) = 0$ , but $f^{''} (x) = 12 x^{2}$ is positive on both sides of $0$ and does not change sign: there is no inflexion. You must check the sign of $f^{''}$ on each side.

Graph of f(x)=x^3-6x^2+9x: the tangents lie above the graph where it is concave down and below it where it is concave up; inflexion at (2, 2)

Second derivative test for extrema

If $f^{'} (x_{0}) = 0$ , the concavity at $x_{0}$ tells you what kind of stationary point it is:

The reason is concavity: a point with a horizontal tangent on a curve that opens upwards is a bottom, and on one that opens downwards it is a peak.

Relationship between the graphs of $f$ , $f^{'}$ and $f^{''}$

Notice that the extrema of $f^{'}$ correspond to points of inflexion of $f$ , because there $f^{''} = 0$ and changes sign.

Graphs of f(x)=x^3-6x^2+9x, of f' and of f'': at x=2, f'' is zero and changes sign, f' has a minimum and f has a point of inflexion

If at $x_{0}$ ...	then the graph of $f$ at $x_{0}$ ...
$f^{'}$ crosses the $x$ -axis	has a horizontal tangent
$f^{'}$ changes from $+$ to $-$	has a local maximum
$f^{'}$ changes from $-$ to $+$	has a local minimum
$f^{'}$ has a maximum or a minimum	has a point of inflexion
$f^{''}$ changes sign	has a point of inflexion

Worked examples

Example 1 Basic

Concavity of a polynomial

Let $f (x) = x^{3} - 6 x^{2} + 2$ .

a) Find $f^{''} (x)$ .

b) Find the coordinates of the point of inflexion.

c) Find the interval on which $f$ is concave up.

Solution

a) $f^{'} (x) = 3 x^{2} - 12 x$ , so $f^{''} (x) = 6 x - 12$ .

b) $f^{''} (x) = 0 \Rightarrow x = 2$ . Since $f^{''}$ is a line with positive gradient, it changes sign at $x = 2$ . Also $f (2) = 8 - 24 + 2 = - 14$ . The point of inflexion is $(2, - 14)$ .

c) $f^{''} (x) > 0$ when $6 x - 12 > 0$ , that is, $x > 2$ . The function is concave up on $] 2, + \infty [$ (and concave down on $] - \infty, 2 [$ ).

Example 2 Medium

When the second derivative test is inconclusive

Let $f (x) = x^{4} - 4 x^{3}$ .

a) Find the stationary points (the values of $x$ with $f^{'} (x) = 0$ ).

b) Use the second derivative to classify the one(s) you can. What happens to the other?

c) Find the points of inflexion of the graph.

Solution

a) $f^{'} (x) = 4 x^{3} - 12 x^{2} = 4 x^{2} (x - 3)$ . It is zero at $x = 0$ and at $x = 3$ .

b) $f^{''} (x) = 12 x^{2} - 24 x = 12 x (x - 2)$ . At $x = 3$ : $f^{''} (3) = 36 > 0$ , so there is a local minimum at $(3, - 27)$ , since $f (3) = 81 - 108 = - 27$ . At $x = 0$ : $f^{''} (0) = 0$ , and the test is inconclusive. Study the sign of $f^{'} = 4 x^{2} (x - 3)$ : it is negative on both sides of $0$ (because $x^{2} > 0$ and $x - 3 < 0$ ). The function decreases before and after, so $x = 0$ is not an extremum.

c) $f^{''} (x) = 12 x (x - 2) = 0$ at $x = 0$ and $x = 2$ . The sign of $f^{''}$ is $+$ for $x < 0$ , $-$ on $] 0, 2 [$ and $+$ for $x > 2$ : it changes at both. The points of inflexion are $(0, 0)$ and $(2, - 16)$ , since $f (2) = 16 - 32 = - 16$ . The one at $x = 0$ has a horizontal tangent.

Example 3 Hard

Finding a cubic from data on its graph

The function $f (x) = x^{3} + a x^{2} + b x + c$ has a point of inflexion at $(1, 0)$ , and the gradient of the tangent at that point is $- 3$ .

a) Find $a$ , $b$ and $c$ .

b) Find the coordinates of the local maximum and of the local minimum.

c) Find the interval on which the graph is concave down.

Solution

a) The data are translated into equations. $f^{'} (x) = 3 x^{2} + 2 a x + b$ and $f^{''} (x) = 6 x + 2 a$ .

Point of inflexion at $x = 1$ : $f^{''} (1) = 6 + 2 a = 0 \Rightarrow a = - 3$ .
Gradient $- 3$ at $x = 1$ : $f^{'} (1) = 3 + 2 a + b = 3 - 6 + b = - 3 \Rightarrow b = 0$ .
The point $(1, 0)$ lies on the graph: $f (1) = 1 - 3 + 0 + c = 0 \Rightarrow c = 2$ .

Then $f (x) = x^{3} - 3 x^{2} + 2$ .

b) $f^{'} (x) = 3 x^{2} - 6 x = 3 x (x - 2)$ , with zeros at $x = 0$ and $x = 2$ . $f^{''} (x) = 6 x - 6$ : $f^{''} (0) = - 6 < 0$ gives a maximum at $(0, f (0)) = (0, 2)$ ; $f^{''} (2) = 6 > 0$ gives a minimum at $(2, f (2)) = (2, - 2)$ .

c) $f^{''} (x) < 0 ⟺ x < 1$ . Concave down on $] - \infty, 1 [$ .

In the exam: when finding parameters, write each equation with its origin ("because it is a point of inflexion", "from the gradient"). For the type of extremum, justify with the sign of $f^{''}$ or of $f^{'}$ . A value without justification does not earn all the marks.

Common mistakes

Summary

$f^{''} > 0$ : concave up ( $f^{'}$ increases). $f^{''} < 0$ : concave down ( $f^{'}$ decreases).
Point of inflexion: $f^{''} (x_{0}) = 0$ and $f^{''}$ changes sign. The extrema of $f^{'}$ correspond to inflexions of $f$ .
With $f^{'} (x_{0}) = 0$ : $f^{''} (x_{0}) > 0$ gives a minimum, $f^{''} (x_{0}) < 0$ gives a maximum, $f^{''} (x_{0}) = 0$ is inconclusive.
To sketch a graph: domain, intercepts, sign of $f^{'}$ , sign of $f^{''}$ , key points and asymptotes.
In Paper 2, the graphs of $f^{'}$ and $f^{''}$ help locate extrema and inflexions.

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Concavity and points of inflexion

Concavity

Points of inflexion

Second derivative test for extrema

Relationship between the graphs of f, f′ and f′′

Worked examples

Common mistakes

Summary

Want to work on it together?

Relationship between the graphs of $f$ , $f^{'}$ and $f^{''}$