Convex Functions: Definition, Properties, Convexity & Solved Examples

A convex function is a type of function where the graph always curves upwards. If you pick any two points on its graph and draw a straight line between them, the line will always lie above or on the graph. That means the function doesn’t dip down in between—it stays smooth and doesn't form valleys.

Some common examples of convex functions are straight lines, U-shaped curves like y=x2 and exponential curves like y=ex.Convex functions are very useful in subjects like math, economics, and data science, especially when solving optimization problems.

In this mathematics article, we will study what is convex functions, strictly convex function, proper convex function, techniques for identifying convexity, and properties of convex functions through worked-out examples.

Convex Function

A real-valued function is considered a convex function in mathematics when the straight line joining any two different points on its graph lies entirely above the function's curve. Alternatively, a function is convex when its epigraph, which is the set of points above or on its graph, is a convex set. 

Additionally, a twice-differentiable function with a single variable is said to be convex if its second derivative is non-negative over its entire domain, and this condition is also necessary for convexity.

Examples of convex functions with a single variable that are commonly known are the quadratic function (), and the exponential function (). Essentially, a convex function is one that takes the form of a cup () in its graph, while a concave function takes the shape of a cap ().

Strictly Convex Function

A function is said to be strictly convex if it satisfies the following condition: for any two distinct points in its domain, the function lies strictly above the line segment that connects the two points. 

In other words, a function is strictly convex if for all and in the domain of , and for all between and ,

,

where is the point on the line segment between and with parameter .

This condition essentially means that the function “curves upwards” in a strictly increasing manner, so that the difference in slopes between two points in the domain increases as the points get further apart. This property is useful in optimization problems, since it implies that there is a unique global minimum for the function.

Proper Convex Function

A function is said to be proper convex if it satisfies the following two conditions:

• The function is convex, which means that for any two points and in its domain and any in the interval , the following inequality holds: .
• The function has no finite values of negative infinity, which means that there does not exist any in its domain such that .

The first condition of convexity ensures that the graph of the function lies above any line segment connecting two points in its domain, while the second condition ensures that the function does not have an arbitrarily large negative value.

Proper convexity is an important property in optimization problems, as it allows for the use of various optimization algorithms, such as gradient descent, to find the minimum of the function. Additionally, many important mathematical tools, such as Legendre-Fenchel duality, are only applicable to proper convex functions.

To learn about types of functions based on set elements with solved examples.

How to Check Convexity of a Function?

To check the convexity of a function, you can use the following methods:

• Second Derivative Test: If the second derivative of a function is always positive or non-negative over an interval, then the function is convex over that interval. Conversely, if the second derivative is always negative or non-positive over an interval, then the function is concave over that interval. If the second derivative changes sign within an interval, then the function is neither convex nor concave over that interval.
• First Derivative Test: If the first derivative of a function is non-decreasing over an interval, then the function is convex over that interval. Conversely, if the first derivative is non-increasing over an interval, then the function is concave over that interval.
• Convexity Properties: Some functions have known convexity properties. For example, any linear function is convex, and any positive power function (e.g., where ) is convex over its domain.

In practice, you can use a combination of these methods to check the convexity of a function over a given interval. If the function is convex, it has some desirable properties, such as having a unique global minimum, which can be useful in optimization problems.

Geometric Interpretation of Convexity

The notion of convexity, which has been introduced, can be easily understood by its simple geometric interpretation. In other words, the concept of convexity can be explained through a straightforward visual representation.

• When a function is convex downward (as depicted in Figure 1), either the midpoint of each chord is located above the corresponding point on the function's graph or it coincides with that point.

• When a function is convex upward (as depicted in Figure 2), the midpoint of each chord is either located below the corresponding point on the function's graph or coincides with that point.

Convex functions possess a notable characteristic that pertains to the position of the tangent on their graph. Specifically, if the function is convex downward on the interval , then its graph won't be positioned below the tangent line drawn at any point within the interval (as depicted in Figure 3).

Similarly, the function is convex upward (or concave downward) on the interval if and only if its graph does not lie above the tangent drawn to it at any point of the segment (as depicted in Figure 4). These characteristics form a theorem that can be demonstrated using the definition of convexity.

To learn about the absolute maxima and minima values with solved examples.

Sufficient Conditions for Convexity

Assuming that the function has a first derivative that exists within a closed interval and a second derivative that exists within an open interval , the subsequent conditions are adequate to determine its convexity or concavity:

• If for all , then the function is convex downward (or concave upward) on the interval .
• If for all , then the function is convex upward (or concave downward) on the interval .

To learn about the concept of interval notation and their types with solved examples.

List of Convex Functions

Here are some common list of convex functions:

• Exponential Functions: A function , is a convex function.
• Quadratic Functions: A function , where is positive, is a convex function.
• Power Functions: A function , where is greater than or equal to , is a convex function.
• Absolute Value Function: A function , is a convex function.
• Exponential Functions with a base less than 1: A function , where is a constant between and , is a convex function.
• Sine and Cosine Functions: A function and are both convex functions.

Convex Function Properties

Assuming that all functions are continuous and defined on the interval , we present a list of properties of convex functions.

• If the functions and are convex downward (upward), then any linear combination where , are positive real numbers is also convex downward (upward).
• If the function is convex downward, and the function is convex downward and non-decreasing, then the composite function is also convex downward.
• If the function is convex upward, and the function is convex downward and non-increasing, then the composite function is also convex downward.
• Any local maximum of a convex upward function defined on the interval is also its global maximum on this interval.
• Any local minimum of a convex downward function defined on the interval is also its global minimum on this interval.

Convex Function Summary

• A function is convex if it lies above its tangent line at every point.
• More formally, a function is convex if for any , in its domain and any between and , we have .
• A function is strictly convex if it lies above its tangent line at every point except for where the tangent line intersects the function.
• Convex functions have no local minima; the global minimum is at the point where the tangent line intersects the -axis.
• Convex functions have second derivatives that are non-negative.
• Linear functions, exponential functions, quadratic functions, and power functions are all examples of convex functions.
• The sum of convex functions is also a convex function.
• Convex functions are useful in optimization problems, as they have unique global minima.

To learn about the derivative rules and differentiation rules in detail with examples. Let us solve some convex function examples to understand the topic.

Operations of Convex Functions

1. Sum of Convex Functions
   If you add two convex functions together, the result is also a convex function.
   Example: If f(x) and g(x) are convex, then f(x)+g(x) is also convex.
2. Multiplication by a Positive Constant
   If you multiply a convex function by a positive number, it remains convex.
   Example: If f(x) is convex and a>0, then a⋅f(x) is also convex.
3. Maximum of Convex Functions
   If you take the maximum of two convex functions at each point, the resulting function is also convex.
   Example: h(x)=max⁡(f(x),g(x)) is convex if both f(x) and g(x) are convex.
4. Composition with a Linear Function
   If you plug a linear function into a convex function, the result is still convex.
   Example: If f(x) is convex and g(x)=ax+b, then f(g(x)) is convex.

Applications of Convex Functions

1. Optimization Problems

Convex functions are widely used in optimization because they make problems easier to solve.

• If a function is convex, any local minimum is also a global minimum.
• This makes finding the best solution (minimum value) much simpler and more reliable.

 2. Economics and Finance

Convex functions appear in many economic models:

• Cost functions and utility functions are often convex.
• They help model consumer behavior, pricing strategies, and profit maximization.

3. Machine Learning & AI

Many learning algorithms are based on minimizing convex loss functions (like Mean Squared Error, Logistic Loss).

• Convex functions ensure that gradient descent converges efficiently.
• It helps train models faster and with more accuracy.

 4. Engineering

In control systems, signal processing, and resource allocation, convex functions are used to design stable and optimal systems.

• Engineers use convex optimization to solve problems like minimizing energy use or maximizing signal strength.

 5. Statistics and Data Analysis

Convex functions are used in:

• Maximum likelihood estimation
• Least squares regression
  These are methods used to find the best-fitting line or model for a set of data points.

Convex Function Solved Examples

1. Find the intervals on which the function f(x) = x³ + ax + b (where a and b are any real numbers) is convex upward.

Solution:
We are given the function:
f(x) = x³ + ax + b

First, find the first derivative:
f'(x) = 3x² + a

Now, find the second derivative:
f''(x) = 6x

To determine convexity, we look at the sign of the second derivative:

• f''(x) > 0 when x > 0 → convex upward
• f''(x) < 0 when x < 0 → concave downward

Therefore, the function is convex upward on the interval (0, ∞).

2.Draw a rough diagram that represents the functions' graphs for all possible combinations of , , , where each value can be either positive or negative.

Solution:

Consider an arbitrary combination of these values, such as the following:

, , .

The function's graph is situated in the upper half-plane, and the function exhibits a strictly decreasing behavior (due to ). Since , the function is convex downward. The figure in the first column and second row provides a diagrammatic representation of this function.

Evidently, there are possible combinations of the three variables with distinct signs. The above figure illustrates the corresponding diagrams of the function graphs.

3.Find the intervals of convexity and concavity of the function .

Solution:

Given function is

Differentiating the function,

Differentiating it again to find the second derivative,

Now the second derivative is equal to zero at the following point:

 

 

The second derivative is positive to the left of this point and negative to the right. Hence, the function is convex downward on and convex upward on .

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