3. (5 points) for the function $f(x)=x^{3}-3x^{2}+5$,\na. (2 points) find all critical points.\nb. (3…
3. (5 points) for the function $f(x)=x^{3}-3x^{2}+5$,\na. (2 points) find all critical points.\nb. (3 points) classify them as local maxima, local minima, or neither. you must use\na sign chart for full credit. *
Answer
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 |
|---|---|---|
| (x<0) | (x=-1) | (f^{\prime}(-1)=3\times(-1)\times(-1 - 2)=9>0) |
| (0<x<2) | (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.