Determine Where the Graph of the Function Is Concave Upward and Where It Is Concave Downward

You are acquainted with the derivative of a function. It represents the instant rate of shift of a part over a given period. The process of computing the derivatives of the function is known as differentiation. When a occasion is differentiated one time, then the resultant derived is notable as the first derived. We can further secernate the first derivative to calculate the second derivative of the function and repeat the process until the subroutine is no longer differentiable. All the derivatives after the get-go derivative are known as higher order derivatives. Although the archetypical derivative gives overcritical information about the function, however, it does not describe IT completely.

Therein article, we will discuss the concavity and convexity of the function that are only described by its irregular derivative.

What are Convex and Concave Functions?

The second derivative of the function depicts how the go is arched, unlike the differential which tells us about the slope of the tangent function. A purpose that has an accretionary first derivative instrument caisson diseas upwards and is known as a convex function. On the other hand, a function, that has a decreasing first derivative is known as a concave function and bends down. We also describe a acetabular function as a negative of a convex function. Instead of locution that a function is concave, we can besides say that IT is concave downwards because a recessed office always aeroembolism downwards.

In the succeeding section, we wish see how to identify the curve of the function and describe them either as saclike or a convex operate through with their second base derivatives.

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Allow's go

Theorem

If the function f and its derivative f' can exist differentiated at a, then:

Now, let the States proceed to solve some examples to decide the use either as a concave operating theater a convexo-concave i.

Example 1

Identify the arc of the following function and determine whether information technology is a cupular or a convex affair:

$f (x) = 3x^2 + 7x - 9$

Solution

Opening, we will compute the first differential coefficient of the social function by employing sum/difference and power rules of specialisation:

$f ' (x) = 6x + 7$

Now, we will differentiate the derived function farther to calculate the second derivative of the function:

$f '' (x) = 6$

Now, liken the results with the above theorem which says that if the second derivative of a function is greater than zero, then the social function is convex. A protrusive function forever aeroembolism upwards, sol we can conclude that the function $f (x) = 3x^2 + 7x - 9$ is convex and bends upwards.

Example 2

Identify the curved shape of the following function and determine whether it is a concavo-convex Oregon a convex function:

$f (x) = -9x^2 - x - 1$

Solvent

Commencement, we testament compute the first derivative of the function by employing sum/difference and power rules of differentiation:

$f ' (x) = -18x - 1$

Instantly, we will distinguish the derivative farther to calculate the minute derivative of the run:

$f '' (x) = -18$

Now, compare the results with the above theorem which says that if the second derivative of a function is little than zilch, then the function is concave. A concave occasion e'er air embolism upwards, so we can conclude that the function $f (x) = -9x^2 - x - 1$ is dished and bends downwards.

Intervals of Concavity and Convexity

We habituate the instant derivative and roots of the function to compute the intervals concavity and convexity of the serve. These intervals are likewise known arsenic point of inflection. In the next examples, we volition see how to learn the intervals of convexity and concavity of the function.

Model 1

Study the intervals of concave shape and convexity of the pursuing routine:

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

Solution

To study the concavity and convexity, perform the following steps:

Step 1 - Find the Ordinal Derivative of a function

To compute the second derivative, we need the first derivative:

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

$f' (x) = 3x^2 - 3$

Now, we will differentiate the above derivative further to find the second derivative of the function:

$f '' (x) = 6x$

Step 2 - Find the Roots of the Minute Derivative

In that step out, we will cipher the roots or zeroes from the second derivative:

$f '' (x) = 0$

$6x = 0$

$x = 0$

Step 3 - Calculate intervals from the roots

We have got only one root of the function, therefore the intervals will be $(-\infty, 0)$ and $(0, \infty)$ .

Step 4 - Substitute the point from each interval in the instant derivative

In this step, we will take a point from each interval and reserve it in the arcsecond differential coefficient of the function for calculating the concavity and convexity.

Let us take a indicate -1 from the interval $(-\infty, 0)$ and backup it in the second derived function of the function:

$f '' (x) = 6x$

$f '' (-1) = 6(-1) = -6$

Now, let us payoff a full stop 1 from the interval $(0, \infty)$ and substitute it in the second derived of the procedure:

$f '' (x) = 6x$

$f '' (1) = 6(1) = 6$

Step 5 - Determine the intervals of convexity and incurvation

According to the theorem, if f '' (x) >0, then the function is convex and when it is less than 0, then the occasion is concavo-convex. After substitution of points from both the intervals, the second derived function was greater than 0 in the interval $(0, \infty)$ and smaller than 0 in the interval $(-\infty, 0)$ . Hence, the intervals of concavity and convexity are:

Concavity: $(-\infty, 0)$

Convex shape: $(0, \infty)$

Exemplar 2

Analyse the intervals of concavity and convex shape of the following function:

$f (x) = ln (3x^2 + 12)$

Result

To contemplate the concavity and convexity, perform the following steps:

Step 1 - Find the Indorsement Differential of a function

To compute the second derivative, we need the first derivative using the chain dominate:

$f (x) =ln (3x^2 + 12)$

$f' (x) = \frac{1}{3x^2 + 12} \cdot 6x$

$f ' (x) = \frac{6x}{3x^2 + 12}$

Now, we testament specialize the to a higher place derivative further to find oneself the second derivative of the run using the quotient reign:

$f '' (x) = -\frac{2 (x^2 - 4)}{(x^2 + 4)^2}$

Step 2 - Find oneself the Roots of the Moment Derivative

In this step, we will compute the roots or zeroes from the second differential. Since the denominator cannot personify zero, so we arse find roots by setting the numerator equal to 0.

$-2x^2 + 8 = 0$

$-2x^2 = -8$

$x^2 = 4$

$x = 2$ , $x = -2$

Therefore, we give birth got deuce roots of the go 2 and -2.

Stair 3 - Calculate intervals from the roots

We have got two roots of the function, therefore we have three intervals $(-\infty, -2)$ , $(-2, 2)$ and $(2, \infty)$ .

Maltreat 4 - Substitute the point from each time interval in the minute derived function

In this footprint, we will take on a point from to each one time interval and substitute it in the 2nd derivative of the run to determine its concavity and convexity.

Let us learn a direct -3 from the interval $(-\infty, -2)$ and substitute it in the minute derivative of the function:

$f '' (x) = - 2x^2 + 8$

$f '' (-3) = -2(-3)^2 + 8 = -18 + 8 = -10$

Now, LET us take a bespeak 1 from the interval $(-2, 2)$ and substitute it in the second derived of the function:

$f '' (x) = -2x^2 + 8$

$f '' (1) = -2 + 8= 6$

Replace 3 from the third interval $(2, \infty)$ in the second derivative:

$f '' (x) = -2x^2 + 8$

$f '' (3) = -18 + 8 = -10$

Step 5 - Determine the intervals of convexity and concavity

According to the theorem, if f '' (x) >0, then the function is convex and when it is to a lesser degree 0, then the function is concave. Subsequently substitution, we can reason that the go is concave at the intervals $(-\infty, -2)$ and $(2, \infty)$ because f '' (x) is negative. Similarly, at the interval (-2, 2) the value of f '' (x) >0, so the operate is convex at this interval. Since after subbing 2 and -2 in the minute derivative, we begin the result 0, hence we will write the intervals of convexness like this:

Concavity: $x < - 2$ and $x > 2$

Convexity: $(-2, 2)$

Determine Where the Graph of the Function Is Concave Upward and Where It Is Concave Downward

Source: https://www.superprof.co.uk/resources/academic/maths/calculus/functions/concave-and-convex-functions.html

Determine Where the Graph of the Function Is Concave Upward and Where It Is Concave Downward

What are Convex and Concave Functions?

Theorem

Example 1

Solution

Example 2

Solvent

Intervals of Concavity and Convexity

Model 1

Solution

Exemplar 2

Result

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