Question

What is the relative maximum of the function?

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Answer 1

Check the picture below.

Answer 2

4

Explanation:

Let
`f(x)` a function and
`x_o` a point of the domain:

The function
`f(x)` presents a relative maximum at
`x_o`, when there exists an environment
`E(x_o)` such that:

`f(x)<f(x_o),\hspace{8}\forall x \in E(x_o),x\\eq x_o`

And the function
`f(x)` presents a relative minimum at
`x_o`, when there exists an environment
`E(x_o)` such that:

`f(x)>f(x_o),\hspace{8}\forall x \in E(x_o),x\\eq x_o`

Necessary condition for the existence of extrema:

Let:

`f: R\rightarrow R`

A function whose domain is
`D=Dom(f)` and
`x_o` a point of the domain:

if
`f` reaches an extreme at
`x_o` and
`f` is differentiable at
`x_o`, then:

`f'(x_o)=0`

If we had the equation of the function we could find the extrema (maxima and minima) mathematically using the previous definition and other criteria. However since we only have the graph, we just can conclude that the maximum of the function is 4 based on this definition:

"The function
`f(x)` presents a relative maximum at
`x_o`, when there exists an environment
`E(x_o)` such that:"

`f(x)<f(x_o),\hspace{8}\forall x \in E(x_o),x\\eq x_o`