Question

Use the graph to state the absolute and local maximum and minimum values of the function.

Short answer

Absolute maximum occurs at \(f(4) = 5\). There is no absolute minimum value. Local maximum occurs at \(f(4) = 5\), \(f(6) = 4\). Local minimum occurs at \(f(2) = 2,f(1) = 3\), \(f(5) = 3\).

Step-by-step solution

Given data

The given function is the graph.

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 maximum occurs at \(f(4) = 5\) because \(f(4) \ge f(x)\) for all value of \(x\) in a domain.

The open dot in the graph indicates that the point is not included in the domain. So, the point \(x = 7\) is not included in the domain. Thus, the minimum value does not occur at the point and hence there is no absolute minimum value.

Since \(f(4) \ge f(x)\) for any \(x\) nearer to 4 in a domain and \(f(6) \ge f(x)\) for any \(x\) nearer to 6 in a domain, local maximum occurs at \(f(4) = 5\) and \(f(6) = 4\).

Since \(f(2) \le f(x)\) for any \(x\) nearer to 2 in a domain, \(f(1) \le f(x)\) for any \(x\) nearer to 1 in a domain, and \(f(5) \le f(x)\) for any \(x\) nearer to 5 in a domain, local minimum occurs at three points such as \(f(2) = 2,f(1) = 3\) and \(f(5) = 3\).