Question

Determine the absolute maximum or minimum, local maximum or minimum, of the given graph.

Short answer

Absolute maximum occurs at $f(4)=5$.

There is no absolute minimum value.

Local maximum occurs at $f(4)=5$ and $f(6)=4$.

Local minimum occurs at $f(2)=2, f(1)=3$ and $f(5)=3$.

Step-by-step solution

Given data

The given function is the number among \(a,b,c,d,r\)and \(s\).

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

The function has absolute maximum at \(x = r\) because \(f(r) \ge f(x)\) in a domain. The function has absolute minimum at \(x = a\) because \(f(a) \le f(x)\) in a domain. The function has local maximum at \(x = b\), \(x = r\) because \(f(b) \ge f(x)\) for any \(x\) nearer to \(b\)and \(f(r) \ge f(x)\) where \(x\) is a point in domain. The function has local minimum at \(x = b\) and \(x = r\) because \(f(b) \le f(x)\) for any \(x\) nearer to \(r\) where \(x\) is a point in domain. The function has local minimum at \(x = d\) because \(f(d) \le f(x)\) for any \(d\) nearer to \(x\) and \(f(r) \le f(x)\) for any \(x\) nearer to \(r\) where \(x\) is a point in domain. There occurs neither maximum nor minimum value at \(x = c\) and \(x = s\).