Question

Let $f(x)=x^3-3 x^2+2 x$. If the equation $f(x)=k$ has exactly one positive and one negative solution then the value of $k$ equals

• A. $-\frac{2 \sqrt{3}}{9}$
• B. $-\frac{2}{9}$
• C. $\frac{2}{3 \sqrt{3}}$
• D. $\frac{1}{3 \sqrt{3}}$

Answer 1

Answer: (A)

$f(x)=x\left(x^2-3 x+2\right)$
$f(x)=x(x-2)(x-1)$
graph of $y=f(x)$ is as shown
now $f ( x )= k$ to have exactly one
positive and negative solution
we have, $k = f \left(1+\frac{1}{\sqrt{3}}\right) $ (think!) $ 1-\frac{1}{\sqrt{3}}$ and $1+\frac{1}{\sqrt{3}}$ are the roots of $f ^{\prime}( x )=0$