Question

Explain the difference between an absolute minimum and a local minimum.

Short answer

The local minimum of \(f(x)\) occurs at \(x = 3\).

Step-by-step solution

Given data

The given function is absolute minimum and local minimum.

Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

Simplify the expression

Absolute minimum: A function \(f\) has an absolute minimum at \(c\) if \(f(c) \le f(x)\quad \forall x\) in the domain \((D)\) of \(f\).

Local minimum: A function \(f\) has local minimum at \(c\) if there is an open interval L (containing \(c\) any \(x\) near to \(c\)) such that \(f(c) \le f(x)\quad \forall x\) in L. Local minimum and local maximum do not attain at extreme values.

Difference between the absolute minimum and the local minimum: From the above, the local minimum is any one of the minimum value available in the curve whereas the absolute minimum is the highest minimum of the curve. That is, the least point of the curve is called as absolute minimum. Therefore, it can be concluded that any absolute minimum is called as local minimum but not the vice versa.

Sketch the graph of the example

Consider an example as shown below in Figure:

From Figure, it is observed that, \(f(1) \le f(x)\quad \forall x\) in \(R\).

Thus, the absolute minimum of \(f(x)\) occurs at \(x = 1\).

Also, it can be observed that, \(f(3) \le f(x)\quad \forall x\) nearer to 3.

Thus, the local minimum of \(f(x)\) occurs at \(x = 3\).