Chapter 3.6, Problem 103PE

(a)

To determine

Let a, b, and c are positive real numbers, where $a<b<c$ and $f\left(x\right)={\left(x-a\right)}^2\left(b-x\right){\left(x-c\right)}^3$. Then the missing values in the sign chart.

The sign chart is as follows:

Sign of ${\left(x-a\right)}^2$.
Sign of $\left(b-x\right)$.
Sign of ${\left(x-c\right)}^3$.
Sign of ${\left(x-a\right)}^2\left(b-x\right){\left(x-c\right)}^3$.

a b c

(b)

To determine

The solution set for $f\left(x\right)>0$. If a, b, and c are positive real numbers, where $a<b<c$ and $f\left(x\right)={\left(x-a\right)}^2\left(b-x\right){\left(x-c\right)}^3$.

Sign of ${\left(x-a\right)}^2$:.	+	+	+
Sign of $\left(b-x\right)$:.	+	$-$	$-$
Sign of ${\left(x-c\right)}^3$: $-$	$-$	$-$	+
Sign of ${\left(x-a\right)}^2\left(b-x\right){\left(x-c\right)}^3$: $-$	$-$	+	$-$

a b c

(c)

To determine

The solution set for $f(x)<0$. If a, b, and c are positive real numbers, where $a<b<c$ and $f\left(x\right)={\left(x-a\right)}^2\left(b-x\right){\left(x-c\right)}^3$.

Sign of ${\left(x-a\right)}^2$:.	+	+	+
Sign of $\left(b-x\right)$:.	+	$-$	$-$
Sign of ${\left(x-c\right)}^3$: $-$	$-$	$-$	+
Sign of ${\left(x-a\right)}^2\left(b-x\right){\left(x-c\right)}^3$: $-$	$-$	+	$-$

a b c