Question

by inspecting the graph of the function, find the absolute maximum and absolute minimum on the given interval.
r(x)=-sqrt3{x}+4 on 1,64
choose the correct graph below.

Explanation:

Step1: Evaluate the function at the endpoints

For \(x = 1\):
\(r(1)=-\sqrt[3]{1}+4=-1 + 4=3\)
For \(x = 64\):
\(r(64)=-\sqrt[3]{64}+4=-4 + 4=0\)

Step2: Analyze the behavior of the function

The function \(y =-\sqrt[3]{x}+4\) is a decreasing function. The cube - root function \(y=\sqrt[3]{x}\) is increasing, and when we multiply it by \(- 1\) and add \(4\), the function \(y =-\sqrt[3]{x}+4\) is decreasing.

Answer:

The absolute maximum is \(3\) (at \(x = 1\)) and the absolute minimum is \(0\) (at \(x = 64\))