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