Question

by inspecting the graph of the function, find the absolute maximum and absolute minimum on the given interval.
r(x)=\sqrt3{x}-3 on 1,64
choose the correct graph below.
a.
b.
c.
d.

Explanation:

Step1: Find the function values at the endpoints

For \(x = 1\):
\(r(1)=\sqrt[3]{1}-3=1 - 3=-2\)
For \(x = 64\):
\(r(64)=\sqrt[3]{64}-3=4 - 3 = 1\)

Step2: Analyze the behavior of the function \(y=\sqrt[3]{x}-3\)

The function \(y = \sqrt[3]{x}\) is an increasing function (its derivative \(y'=\frac{1}{3x^{\frac{2}{3}}}>0\) for \(x
eq0\)). When we subtract 3 (\(y=\sqrt[3]{x}-3\)), the shape of the graph (increasing nature) remains the same.

Answer:

The absolute minimum is \(-2\) (at \(x = 1\)) and the absolute maximum is \(1\) (at \(x = 64\)).