Question

for the graph of the function \\(y = f(x)\\), find the absolute maximum and the absolute minimum, if it exists.

Explanation:

Identify the given points and domain

Using the Graph Analysis knowledge point
The function \(y = f(x)\) is defined on the closed interval \([0, 7]\). The key coordinates plotted on the graph are:

• Left endpoint: \((0, 2)\)
• Local maximum: \((3, 6)\)
• Local minimum: \((5, 1)\)
• Right endpoint: \((7, 4)\)

Determine the absolute maximum

Using the Local Extrema and Graph Analysis knowledge points
The absolute maximum is the largest \(y\)-value that the function achieves on its entire domain.
Comparing the \(y\)-values at all critical points and endpoints:

• At \(x = 0\), \(y = 2\)
• At \(x = 3\), \(y = 6\)
• At \(x = 5\), \(y = 1\)
• At \(x = 7\), \(y = 4\)

The largest \(y\)-value is \(6\), which occurs at \(x = 3\).

Determine the absolute minimum

Using the Local Extrema and Graph Analysis knowledge points
The absolute minimum is the smallest \(y\)-value that the function achieves on its entire domain.
Comparing the \(y\)-values at all critical points and endpoints:

• At \(x = 0\), \(y = 2\)
• At \(x = 3\), \(y = 6\)
• At \(x = 5\), \(y = 1\)
• At \(x = 7\), \(y = 4\)

The smallest \(y\)-value is \(1\), which occurs at \(x = 5\).

Answer:

Absolute maximum is \(6\) at \(x = 3\).
Absolute minimum is \(1\) at \(x = 5\).