Question

If $f\left(x\right)=x^5-5x^4+5x^3-10$ has local maxima and minima at $x=a$ and $x=b$ respectively, then $(a,b)$ is

• A. (0,1)
• B. (1,3)
• C. (1,0)
• D. (3,0)

Answer 1

The correct option is B (1,3)

$f\left(x\right)=x^5-5x^4+5x^3-10$
$f^{\prime }\left(x\right)=5x^4-20x^3+15x^2$
For maxima or minima,
$f^{\prime }\left(x\right)=0$
$\Rightarrow 5x^2(x-1)(x-3)=0$
$\Rightarrow x=0,1,3$
$f^{\prime \prime }\left(x\right)=20x^3-60x^2+30x$
$f^{\prime \prime }\left(x\right)<0$ at $x=1$
Hence f(x) has a local maximum at $x=1$
$f^{\prime \prime }\left(x\right)>0$ t $x=3$
Hence f(x) has a local minimum at $x=3$.
So, $a=1,b=3$