Question

Sketch the graph of a function \({\bf{f}}\)that is continuous on \(\left( {{\bf{1}},{\bf{5}}} \right)\) and has the given properties.

Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4.

Short answer

The graph of a function \(f\)that is continuous on \(\left( {1,5} \right)\) is shown below:

Step-by-step solution

Definition for an absolute minimum and maximum

For any number \(c\) in the domain \(D\)of a function \(f\), \(f\left( c \right)\) is said to be the absolute minimum value of \(f\) on \(D\) if \(f\left( c \right) \le f\left( x \right)\) for all \(x\) in \(D\).

For any number \(c\) in the domain \(D\) of a function \(f\), \(f\left( c \right)\) is said to be the absolute maximum value of \(f\) on \(D\) if \(f\left( c \right) \ge f\left( x \right)\) for all \(x\) in \(D\).

Definition for a local minimum and maximum

Any number \(c\) in the domain \(D\) of a function \(f\), \(f\left( c \right)\) is said to be the local minimum value of \(f\) if \(f\left( c \right) \le f\left( x \right)\) when \(x\) is near \(c\).

For any number \(c\) in the domain \(D\) of a function \(f\), \(f\left( c \right)\) is said to be the local maximum value of \(f\) if \(f\left( c \right) \ge f\left( x \right)\) when \(x\) is near \(c\).

Describe the given properties to graph the function

Given that the function \(f\) is continuous on \(\left( {1,5} \right)\). This means the graph has no breakpoints in the given interval \(\left( {1,5} \right)\).

The graph of \(f\) has a local maximum at \(x = 3\), so, the graph of \(f\)changes from increasing to decreasing at the point \(x = 3\).

The graph of \(f\) has a local minimum at \(x = 2\), so, the graph of \(f\) changes from decreasing to increasing at the point \(x = 2\).

The graph of \(f\)has a local minimum at \(x = 4\), so, the graph of \(f\) changes from decreasing to increasing at the point \(x = 4\).

The absolute maximum is at \(x = 5\), so, the function \(f\) has greatest possible value at \(x = 5\).

The absolute minimum is at \(x = 4\), so, the function \(f\) has the smallest possible value at \(x = 4\).

Sketch the graph of the function

The graph of a function \(f\)that is continuous on \(\left( {1,5} \right)\) is shown below: