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)

$\frac{d y}{d x}=3 x^2+2 x-1=0 ; 3 x^2+3 x-x-1=0$
$3 x(x+1)-(x+1)=0 \Rightarrow x=-1 \text { or } x=1 / 3$
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 $\quad y=1$ and $y=-\frac{5}{27}$
distance between the tangents $1+\frac{5}{27}=\frac{32}{27}$