Question

use the graph of the given function to find any relative maxima and relative minima.
f(x)=x^{3}-12x + 2
a. minimum: (2,-14)
maximum: (-2,18)
b. maximum: (2,-14)
minimum: (-2,18)
c. maxima: (-2,18) and (0,0)
minimum: (2,-14)
d. no maxima or minima

Explanation:

Step1: Identify relative maxima and minima from graph

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.
From the graph of \(y = f(x)=x^{3}-12x + 2\), we observe the following:

• At \(x=-2\), the function changes from increasing to decreasing. The \(y\) - value at \(x =-2\) is \(y=f(-2)=(-2)^{3}-12\times(-2)+2=-8 + 24+2=18\). So, the relative maximum is \((-2,18)\).
• At \(x = 2\), the function changes from decreasing to increasing. The \(y\) - value at \(x=2\) is \(y=f(2)=2^{3}-12\times2+2=8-24 + 2=-14\). So, the relative minimum is \((2,-14)\).

Answer:

A. Minimum: \((2,-14)\) Maximum: \((-2,18)\)