Question

The maxima and minima of the function $f\left(x\right)=2x^3-15x^2+36x+10$ occur, respectively at

• A. $x=3$ and $x=2$
• B. $x=1$ and $x=3$
• C. $x=2$ and $x=3$
• D. $x=3$ and $x=4$

Answer 1

The correct option is C $x=2$ and $x=3$

$f\left(x\right)=2x^3-15x^2+36x+10$
$f^{\prime }\left(x\right)=6x^2-30x+36$
$=6(x^2-5x+6)$
$f^{\prime }\left(x\right)=6(x-2)(x-3)$
For critical points:
$f^{\prime }\left(x\right)=0$
$6(x-2)(x-3)=0$
$x=2,3$
$f^{\prime \prime }\left(x\right)=12x-30$
$f^{\prime \prime }\left(2\right)=24-30<0$
$\Rightarrow f\left(x\right)$ has local maxima at $x=2$
$f^{\prime \prime }\left(3\right)=12\left(3\right)-30=6>0$
$\Rightarrow f\left(x\right)$ has local minima at $x=3$