Question

The function $f(x)=x^5-5x^4+5x^3-1$ has

• A. one minima and two maxima
• B. two minima and one maxima
• C. two minima and two maxima
• D. one minima and one maxima

Answer 1

Answer: (D)

$f(x)=5x^4-20x^3+15x^2$
$f^{\prime \prime }(x)=20x^3-60x^2+30x$
$f^{\prime }(x)=0$
$\Rightarrow 5x^2(x^2-4x+3)=0$
$\Rightarrow x=0$, $1$, $3$
$f^{\prime \prime }(1)=-10<0$ and
$f^{\prime \prime }(3)=540-540+90=90>0$
$f^{\prime \prime }(0)=0$ and $f^{\prime \prime \prime }(0)\ne 0$
So, $x=0$ is a point of inflexion.
$\therefore$ maximum at $x=1$,
minimum at $x=3$.