Question

1. (5 points) for the function $f(x)=x^{3}-3x^{2}+5$,

a. (2 points) find all critical points.
b. (3 points) classify them as local maxima, local minima, or neither. you must use
a sign chart for full credit. *

Explanation:

Step1: Find the derivative

The derivative of \(f(x)=x^{3}-3x^{2}+5\) is \(f^{\prime}(x)=3x^{2}-6x = 3x(x - 2)\).

Step2: Find critical points

Set \(f^{\prime}(x)=0\), so \(3x(x - 2)=0\). Solving gives \(x = 0\) and \(x=2\).

Step3: Create a sign chart

Interval	Test Value	\(f^{\prime}(x)\) Sign
\(0	\(x = 1\)	\(f^{\prime}(1)=3\times1\times(1 - 2)=-3<0\)
\(x>2\)	\(x=3\)	\(f^{\prime}(3)=3\times3\times(3 - 2)=9>0\)

Step4: Classify critical points

Since \(f^{\prime}(x)\) changes from positive to negative at \(x = 0\), \(x = 0\) is a local maximum.
Since \(f^{\prime}(x)\) changes from negative to positive at \(x = 2\), \(x = 2\) is a local minimum.

Answer:

a. Critical points are \(x = 0\) and \(x=2\).
b. \(x = 0\) is a local maximum and \(x = 2\) is a local minimum.