Question

(sections 2.1,2.2,2.6)
11 of 17
this q
this q
use the graph of the given function to find any relative maxima and relative
minima.
f(x)=x³ - 3x² + 1

Explanation:

Step1: Recall the definition of relative maxima and minima

A relative maximum is a point where the function changes from increasing to decreasing. A relative minimum is a point where the function changes from decreasing to increasing.

Step2: Analyze the graph

Looking at the graph of \(y = f(x)=x^{3}-3x^{2}+1\), we observe the turning points.

At \(x = 0\), the function changes from increasing to decreasing. So, we find \(f(0)=(0)^{3}-3(0)^{2}+1 = 1\).

At \(x = 2\), the function changes from decreasing to increasing. So, we find \(f(2)=(2)^{3}-3(2)^{2}+1=8 - 12 + 1=-3\).

Answer:

The relative maximum is \((0,1)\) and the relative minimum is \((2,-3)\)