The extrema of a function is a crucial characteristic to observe for math functions. This post explains the basics of it:
Extrema: Minima and Maxima
Most function graphs will have extrema (singular: extremum). These are the maximum and minimum points along its curve.
- The maximum is the point along the graph at the "top of the hill."
- The minimum is the point along the graph at the "bottom of the valley."
For example, in the sine graph \(f(x)=\sin x\), we say that at 90 degrees, located at \((\frac{\pi}{2},1)\), is the maximum point of the graph (since it is the apex point of the function). The point at \((\frac{3\pi}{2},-1)\) is the minimum point of the graph.
Let's formalize the definition of maximum and minimum. Consider a graph of a function \(f\) with an interval, either closed or open. In addition, let's have a number \(c\) between said interval.
- If \(f(c) \leq f(x)\) for all values of the domain, then \(f(c)\) is the minimum of the function.
- If \(f(c) \geq f(x)\) for all values of the domain, then \(f(c)\) is the maximum of the function.
At points of extrema, the slope at these points is equal to zero; hence, the first derivative must be zero. If the first derivative is impossible to obtain (like for sharp-pointed graphs), it is also an extremum.
Let's note that not all functions have an extrema. An example of which is a horizontal line.
Absolute and Relative
There are two types of extrema: absolute and relative.
- An extrema is absolute if the value of \(f(c)\) is the highest/lowest for all possible \(f(x)\).
- An extrema is relative if \(f(c)\) is the highest/lowest for a specific interval.
To illustrate the difference, let's consider a sample graph of a function and make several observations:
- Four extrema are located at points \(c_1\), \(c_2\), \(c_3\), and \(c_4\).
- Extrema \(c_1\) and \(c_3\) are maximum while \(c_2\) and \(c_4\) are minimum.
- Between \(c_1\) and \(c_3\), \(c_1\) is the absolute maximum because it is the highest point among all other values.
- Between \(c_2\) and \(c_4\), \(c_4\) is the absolute minimum because it is the lowest point among all other values.
- When we consider an interval along the function, say the shaded interval, point \(c_2\) is a relative minimum in the interval.
- If we consider the same interval, point \(c_3\) is a relative maximum.
Critical Numbers
The x-value of the extremum is called the critical number \(c\). From this, we can designate the points of maximum and minimum as coordinate \((c, f(c))\). These values are essential when it comes to many applications of differentiation, such as optimization.
Summary
Most function graphs will have extrema (singular: extremum). These are the maximum and minimum points along its curve.
Consider a graph of a function \(f\) with an interval, either closed or open. In addition, let's have a number \(c\) between said interval. If \(f(c) \leq f(x)\) for all values of the domain, then \(f(c)\) is the minimum of the function. If \(f(c) \geq f(x)\) for all values of the domain, then \(f(c)\) is the maximum of the function.
At points of extrema, the slope at these points is equal to zero; hence, the first derivative must be zero. If the first derivative is impossible to obtain (like for sharp-pointed graphs), it is also an extremum.
There are two types of extrema: absolute and relative.
An extrema is absolute if the value of \(f(c)\) is the highest/lowest for all possible \(f(x)\).
An extrema is relative if \(f(c)\) is the highest/lowest for a specific interval.
The x-value of the extremum is called the critical number \(c\). From this, we can designate the points of maximum and minimum as coordinate \((c, f(c))\).
