Question

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

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

Answer 1

The correct option is C $(1,3)$

$f\left(x\right)=x^5-5x^4+5x^3-10$
$\Rightarrow f^{\prime }\left(x\right)=5x^4-20x^3+15x^2$
From $f^{\prime }\left(x\right)=0$
$\Rightarrow 5x^2(x^2-4x+3)=0$
$\Rightarrow 5x^2(x-3)(x-1)$
$\Rightarrow x=0,1,3$

From above sign change at $x=1,f\left(x\right)$ has local maximum and at $x=3,f\left(x\right)$ has local minimum
Hence, $(a,b)=(1,3)$