What are the Extreme Points of a Function?

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

What are Maxima and Minima?

The maxima and minima of a function, generally known as extrema in mathematical analysis, are the function's greatest and lowest values, either within a specific range or throughout the whole domain.

Maxima of a Function

The maxima is the capability that a function has a maximum value at a point x=a.

If $f(x)\le f(a)$for all $x\in (a-h,a+h)$, where h is somewhat small but positive quantity, then the point $x=a$ is named as a point of maximum of function $f(x)$, and $f(a)$, as shown in the graph, is known as the maximum value or the greatest value of the function. Consider the below graph for the maxima of a function.

Minima of a Function

The function $y=f(x)$ is said to have a regional minimum at a point $x=a$, if$f(x)\ge f(a)$ for all $x\in (a-h,a+h)$, where h is somewhere small but positive quantity. The point is referred to as a point of minimum of function $f(x)$, and $f(a)$ is known as the minimum value or the least value of $f(x)$. Consider the below graph for the minima of a function.

Properties of Maxima and Minima

Maximum and Minimum Values in a Closed Interval

Let $f(x)$ be a function on [a, b] and $c$ be a point in the interval [a, b].

1. If for each point $x$ in $[a,b],f(x)\ge f(c)$, then $f(c)$ is the absolute (or global) minimum value.

2. Similarly, if for each point $x$ in $[a,b],f(x)\le f(c)$, then $f(c)$ is the absolute (or global) maximum value.

3. If for each point $x$ in the closed interval $[x_1,x_2]$, $f(x)\ge f(c)$, then $f(c)$ is the local minimum value. Here, $c$ is a point in the interval $[x_1,x_2]\subset [a,b]$.

4. Similarly, if for each point $x$ in closed interval $[x_1,x_2]$, $f(x)\le f(c)$, then $f(c)$ is the local maximum value. Here, $c$ is a point in the interval $[x_1,x_2]\subset [a,b]$.

5. If $f(x)$ is continuous on [a, b] and differentiable in $(a,b)$, a point $c$ in [a, b] is called a critical point of $f(x)$, if either $f\text{'}(c)$ does not exist, or $f\text{'}(c)=0$. i.e. derivative will be zero.

Local Maximum

A function $f(x)$ is also said to have attained a local maximum at $x=a$, if there exists a neighborhood $(a-\delta ,a+\delta )$ of $a$ such that,

In such case$f(a)$ is said to attain a local maximum value $f(x)$ at $x=a$.

Local Minimum 

A function $f(x)$ is also said to have attained a local minimum at $x=a$, if there exists a neighborhood $(a-\delta ,a+\delta )$ of $a$ such that,

In such a case$f(a)$ is called the local minimum value of $f(x)$ at $x=a$.

Methods to Find Local Extremum

1. First Derivative Test

Let $f(x)$ be a differentiable function on an interval $I$, and $a\in I$. Then:

A. Point a is local maximum of $f(x)$ if:

(a) $f\text{'}(a)=0$

(b) $f\text{'}(x)>0$ for $x\in (a-h,a)$and$f\text{'}(x)<0$ for $x\in \left(a,a+h\right)$,where h is a small but positive quantity.

B. Point a is the local minimum of $f\left(x\right)$, if

(a) $f\text{'}(a)=0$

(b) $f\text{'}(x)<0$ for $x\in (a-h,a)$and$f\text{'}(x)>0$ for $x\in \left(a,a+h\right)$,where h is a small positive quantity.

C. If $f^{\text{'}}\left(a\right)=0$ i.e derivative is equal to zero but $f\text{'}\left(x\right)$ does not alter the sign in the interval $\left(a-h,a+h\right)$, for any positive quantity h, then x=a is neither a point of minimum nor a point of maximum.

2. Second Derivative Test

Let $f\left(x\right)$ be a differentiable function on an interval $I$. Let $a\in I$be such that the derivative is continuous at$x=a$, then:

A. $x=a$ is a local maximum point, if$f^{\text{'}}\left(a\right)=0$and,$f^{\text{'}\text{'}}\left(a\right)<0$ .

B. $x=a$ is a local minimum point, if $f^{\text{'}}\left(a\right)=0$and $f^{\text{'}\text{'}}\left(a\right)>0$.

C. If $f^{\text{'}}\left(a\right)=f^{\text{'}\text{'}}\left(a\right)=0$ , then $x=a$is called point of inflection.

D. If $f^{\text{'}}\left(a\right)=f^{\text{'}\text{'}}\left(a\right)=f^{\text{'}\text{'}\text{'}}\left(a\right)=0$ and $f^4\left(a\right)<0$, then x=a is a point of local maximum, and if $f^4>0$, then x=a is a point of local minimum.

Global Maxima

Consider a function f(x) and a point c lying in the domain of the function f.

When there are no other points in the domain of the function where the value of the function is greater than the value of the function at x=c, i.e. f(c), the point is known as a global maximum.

Types of Global Maxima:

• Global maxima can satisfy all of the conditions of local maxima. You can also understand it as the local maxima with the highest value in this case.
• Alternately, the global maxima for an increasing function could be the endpoint in its domain; as it would obviously have the extreme value.

Common Mistakes

1. To find the value of the local highest or lowest point, substitute the value of x in the 2nd derivative, $f\text{'}\text{'}\left(x\right)$, not in 1st derivative $f\text{'}\left(x\right)$.

2. The final common mistake is to confuse extrema with extreme values by writing, for instance, “$f\left(x\right)=x^2+1$ has the minimum f(0)=1“. The point x=0 is indeed a local minimum of $f\left(x\right)=x^2+1$, however, the value f(0)=1 is called a local minimum value, not a local minimum point.

Context and Applications

This topic is significant in the professional exams for undergraduate and postgraduate courses, especially for:

1. B.Sc. Mathematics
2. M.Sc. Mathematics

Related Concepts

1. Differentiation (derivative) from calculus.
2. Double differentiation from calculus.
3. Tangent

Practice Problem

1. Find the (1) maxima and (2) minima for:

$y=5x^3 +2x^2 -3x$

Differentiate the function:

$\frac{dy}{dx}\text{= }15x^2 +4x-3$

The derivative is quadratic with zeros at:

Now, the second derivative is $y\text{'}\text{'}=30x+4$.

At $x=-3/5$;

$y"<0$ , it means −3/5 is a point of local maximum.

At $x=+1/3:$

$y\text{'}\text{'}>0$, it means +1/3 is a point of local minimum.

Example 2: What is the worth of $\left(\text{x - 1}\right){\left(\text{x - 2}\right)}^{\text{2}}$ at its maxima?

Solution:

Given

Differentiate the function and evaluate the critical points:

Differentiate the function again and check the sign of the second derivative at the critical points:

Hence, at $x=4/3$ is the point of local maximum.

∴ $Maximumvalue=f\left(4/3\right)=4/27$

Example 3: What is the supreme value of $x^{3 }-12x^2 +36x+17$in the interval $\left[1,10\right].$solve:

Let $f\left(x\right)=x^{3 }-12x^2 +36x+17$

∴ $f\prime \left(x\right)=3x^{2 }-24x+36=0atx=2,6$

Now $f\prime \prime \left(x\right)=6x-24$ is negative at $x=2$.

Also $f\left(6\right)=17,f\left(2\right)=49$.

At the endpoints $f\left(1\right)=42,f\left(10\right)=177$.

Hence, $f\left(x\right)$has its supreme value as $177$.

2. How do you find the absolute maxima?

Answer:

Initially, catch on all the critical numbers within the given interval. Then, plug in every single critical number from the first step into the function i.e. f(x).

Plug the endpoints of the given interval into f(x). The highest value is the absolute maxima and the lowest value is the absolute minima.

3. What are the global maxima and global minima?

Answer: In calculus, the global maxima of any function is identified as the global greatest of that function. The global minimum of any function is identified as the global least of that function.