Question

Find all the points of local maxima and local minima of the function $f(x)=(x-1{)}^3(x+1{)}^2$

• A. $1$, $-1$, $-1/5$
• B. $1$, $-1$
• C. $1$, $-1/5$
• D. $-1$, $-1/5$

Answer 1

Answer: (A)

Let $y=f(x)=(x-1{)}^3(x+1{)}^2$. Then,
$\frac{dy}{dx}=3{\left(x-1\right)}^2{\left(x+1\right)}^2+2\left(x+1\right){\left(x-1\right)}^3$
$\Rightarrow \frac{dy}{dx}={\left(x-1\right)}^2\left(x+1\right)\left\{3\left(x+1\right)+2\left(x-1\right)\right\}$
$\Rightarrow \frac{dy}{dx}={\left(x-1\right)}^2\left(x+1\right)\left(5x+1\right)$
For local maximum or local minimum, we have
$\frac{dy}{dx}=0\Rightarrow {\left(x-1\right)}^2\left(x+1\right)\left(5x+1\right)=0$
$\Rightarrow x=1$ or, $x=-1$ or,
$x=-\frac{1}{5}$