Question

The function $x^5-5x^4+5x^3-1$ is

• A. maximum at $x=3$ and minimum at $x=1$
• B. minimum at $x=1$
• C. neither maximum nor minimum at $x=0$
• D. maximum at $x=0$

Answer 1

Answer: (C)

Let, $f\left(x\right)=x^5-5x^4+5x^3-1$
$\Rightarrow f^{{}^{\prime }}\left(x\right)=5x^4-20x^3+15x^2=0$
$\therefore x^2$ $\left(x-3\right)\left(x-1\right)=0$ or $x=3,1,0$
Now, $f^{{}^{\prime \prime }}\left(x\right)=20x^3-60x^2+30x$
Put $x=3,1,0$ , we get $f^{{}^{\prime \prime }}\left(3\right)>0,f^{{}^{\prime \prime }}\left(1\right)<0,f^{{}^{\prime \prime }}\left(0\right)=0$
Hence $f\left(x\right)$ is minimum at $x=3,$ maximum at $x=1$ , neither maximum nor minimum at $x=0.$