Question

Let $f\left(x\right)=(x-1{)}^4\cdot (x-2{)}^n,nϵN.$ Then $f\left(x\right)$ has

• A. a maximum at $x=1$ if $n$ is odd
• B. a maximum at $x=1$ if $n$ is even
• C. a minimum at $x=2$ if $n$ is even
• D. a maximum at $x=2$ if $n$ is odd

Answer 1

Answer: (A), (C)

a maximum at $x=1$ if $n$ is odd
a minimum at $x=2$ if $n$ is even

Considering the nth Derivative test,
If n is odd,
We get a maximum at $(x-1{)}^4=0$
Or
$x=1$ ...(i)
If n is even, we get a minimum at $(x-2{)}^n=0$
Or
$x=2$ ...(ii)