Question

If $f\left(x\right)=(x-1{)}^2(x+1{)}^2$, then the function $f$ has

• A. $\text{a local maximum at}x=\frac{1}{5}$
• B. $\text{a local minimum at}x=\frac{1}{5}$
• C. $\text{a local minimum at}x=-1$
• D. $\text{a local maximum at}x=-1$

Answer 1

The correct option is A $\text{a local maximum at}x=\frac{1}{5}$

$f\left(x\right)=(x-1{)}^2(x+1{)}^3$
$\Rightarrow f^{\prime }\left(x\right)=2(x-1)(x+1{)}^3+3(x+1{)}^2(x-1{)}^2=0\Rightarrow (x-1\left)\right(x+1{)}^2(5x-1)=0\Rightarrow x=-1,\frac{1}{5},1$
$\Rightarrow f\left(x\right)$ has local maxima at $\frac{1}{5}$