Question

The curve $y=x^3+x^2-x$ has two horizontal tangents. The distance between these two horizontal lines, is

• A. $\frac{13}{9}$
• B. $\frac{11}{9}$
• C. $\frac{22}{27}$
• D. $\frac{32}{27}$

Answer 1

Answer: (D)

$\frac{dy}{dx}=3x^2+2x-1=0;3x^2+3x-x-1=0$
$3x(x+1)-(x+1)=0\Rightarrow x=-1\text{ or }x=1/3$
when $x=-1;y=1$
$x=1/3;y=\frac{1}{27}+\frac{1}{9}-\frac{1}{3}=\frac{1+3-9}{27}=-\frac{5}{27}$
Two tangents are $y=1$ and $y=-\frac{5}{27}$
distance between the tangents $1+\frac{5}{27}=\frac{32}{27}$